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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>;<br/>

<b>thus </b><a NAME="E1:56"/>
( <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> implies  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E53">L1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1"/>
( <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> or <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> or <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> )
 </span><b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a>, <a class="txt" href="rinfsup2.html#E1:56_1"><i><font color="Green" title="E53">L1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1"/><i><font color="Green" title="E54">C1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:56_1_1_1"/><i><font color="Green" title="E55">C11</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c2">k</font> is <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T19" target="_self" title="RINFSUP2:th.19">Lm3002y</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">rseq0</font> = <font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c2">k</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Lm300a</a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_1"><i><font color="Green" title="E55">C11</font></i></a></i>;<br/><a NAME="E5:56_1_1_1"/>
<font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c2">k</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T20" target="_self" title="RINFSUP2:th.20">Lm3002x</a>, <a class="txt" href="rinfsup2.html#E1:56_1_1_1"><i><font color="Green" title="E54">C1</font></i></a></i>;<br/><a NAME="E6:56_1_1_1"/><b>then </b><i><font color="Green" title="E56">C13z</font></i>: 
<font color="Maroon" title="c3">rseq0</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T15" target="_self" title="RINFSUP2:th.15">Lm300g</a></i>;<br/><a NAME="E7:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">rseq0</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T90" title="RINFSUP1:th.90">RINFSUP1:90</a></i>;<br/><a NAME="E8:56_1_1_1"/><b>then </b>
(  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c2">k</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c2">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T9" target="_self" title="RINFSUP2:th.9">Lm202</a>, <a class="ref" href="rinfsup2.html#T10" target="_self" title="RINFSUP2:th.10">Lm203</a></i>;<br/><a NAME="E9:56_1_1_1"/><b>then </b>
(  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T28" target="_self" title="RINFSUP2:th.28">Lm203a</a>, <a class="ref" href="rinfsup2.html#T29" target="_self" title="RINFSUP2:th.29">Lm203b</a></i>;<br/><b>hence </b><a NAME="E10:56_1_1_1"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E6:56_1_1_1"><i><font color="Green" title="E56">C13z</font></i></a>, <a class="ref" href="rinfsup1.html#T90" title="RINFSUP1:th.90">RINFSUP1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_2"/><i><font color="Green" title="E54">C2</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">g</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:56_1_1_2_1"/>
<font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:56_1_1_2_1"/><i><font color="Green" title="E55">C22</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a>, <a class="txt" href="rinfsup2.html#E1:56_1_1_2"><i><font color="Green" title="E54">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:56_1_1_2_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/><a NAME="E2:56_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E56">C24</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E3:56_1_1_2_1"><i><font color="Green" title="E55">C22</font></i></a></i>;<br/><a NAME="E3:56_1_1_2_1_1"/>
<font color="Maroon" title="c4">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:56_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T8" target="_self" title="RINFSUP2:th.8">LmXXX</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="rinfsup2.html#E2:56_1_1_2_1_1"><i><font color="Green" title="E56">C24</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:56_1_1_2_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:56_1_1_2"/><b>then </b><i><font color="Green" title="E55">D1</font></i>: 
 <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a></i>;<br/><a NAME="E4:56_1_1_2"/><b>then </b>
 <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E5:56_1_1_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_2"><i><font color="Green" title="E55">D1</font></i></a></i>;<br/><a NAME="E6:56_1_1_2"/><b>then </b><i><font color="Green" title="E56">S1</font></i>: 
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="rinfsup2.html#T37" target="_self" title="RINFSUP2:th.37">Lm310b</a></i>;<br/><a NAME="E7:56_1_1_2"/><i><font color="Green" title="E57">D2a</font></i>: 
 <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c2">iseq</font> =  <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font>;<br/>
<b>assume </b><a NAME="E1:56_1_1_2_2"/>
not  <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:56_1_1_2_2"/><i><font color="Green" title="E58">P11</font></i>: 
( <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> ) ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:56_1_1_2_2"/><i><font color="Green" title="E59">P12</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a>, <a class="txt" href="rinfsup2.html#E1:56_1_1_2"><i><font color="Green" title="E54">C2</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_2_2"><i><font color="Green" title="E58">P11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:56_1_1_2_2_1"/>
<font color="Maroon" title="c5">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">Y</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E3:56_1_1_2_2_1"/><i><font color="Green" title="E60">P132</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D7" target="_self" title="RINFSUP2:def.7">Defsuperior</a></i>;<br/><b>assume </b><a NAME="E4:56_1_1_2_2_1"/><i><font color="Green" title="E61">P131</font></i>: 
<font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:56_1_1_2_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:56_1_1_2_2_1_1"/><i><font color="Green" title="E62">W2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="rinfsup2.html#E3:56_1_1_2_2_1"><i><font color="Green" title="E60">P132</font></i></a></i>;<br/><a NAME="E4:56_1_1_2_2_1_1"/>
<font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E4:56_1_1_2_2_1"><i><font color="Green" title="E61">P131</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_2_2_1_1"><i><font color="Green" title="E62">W2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_2_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E5:56_1_1_2_2"><i><font color="Green" title="E59">P12</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_2_2_1_1"><i><font color="Green" title="E62">W2</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:56_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/><b>hence </b><a NAME="E7:56_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_2_2_1"><i><font color="Green" title="E60">P132</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rinfsup2.html#E3:56_1_1_2_2"><i><font color="Green" title="E58">P11</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:56_1_1_2"/><b>then </b>
 <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E9:56_1_1_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a>, <a class="txt" href="rinfsup2.html#E7:56_1_1_2"><i><font color="Green" title="E57">D2a</font></i></a></i>;<br/><b>hence </b><a NAME="E10:56_1_1_2"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T36" target="_self" title="RINFSUP2:th.36">Lm310a</a>, <a class="txt" href="rinfsup2.html#E6:56_1_1_2"><i><font color="Green" title="E56">S1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3"><b>suppose </b></a><a NAME="E1:56_1_1_3"/><i><font color="Green" title="E54">C3</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">g</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:56_1_1_3_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">g</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:56_1_1_3_1"/><i><font color="Green" title="E55">C22</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a>, <a class="txt" href="rinfsup2.html#E1:56_1_1_3"><i><font color="Green" title="E54">C3</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:56_1_1_3_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/><a NAME="E2:56_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E56">C24</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E3:56_1_1_3_1"><i><font color="Green" title="E55">C22</font></i></a></i>;<br/><a NAME="E3:56_1_1_3_1_1"/>
<font color="Maroon" title="c4">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:56_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T8" target="_self" title="RINFSUP2:th.8">LmXXX</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_3_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="rinfsup2.html#E2:56_1_1_3_1_1"><i><font color="Green" title="E56">C24</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:56_1_1_3_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:56_1_1_3"/><b>then </b><i><font color="Green" title="E55">D1</font></i>: 
 <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></i>;<br/><a NAME="E4:56_1_1_3"/><b>then </b>
 <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E5:56_1_1_3"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_3"><i><font color="Green" title="E55">D1</font></i></a></i>;<br/><a NAME="E6:56_1_1_3"/><b>then </b><i><font color="Green" title="E56">S1</font></i>: 
 <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="rinfsup2.html#T36" target="_self" title="RINFSUP2:th.36">Lm310a</a></i>;<br/><a NAME="E7:56_1_1_3"/><i><font color="Green" title="E57">D2a</font></i>: 
 <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c2">iseq</font> =  <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font>;<br/>
<b>assume </b><a NAME="E1:56_1_1_3_2"/>
not  <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:56_1_1_3_2"/><i><font color="Green" title="E58">P11</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> ) ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:56_1_1_3_2"/><i><font color="Green" title="E59">P12</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a>, <a class="txt" href="rinfsup2.html#E1:56_1_1_3"><i><font color="Green" title="E54">C3</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_3_2"><i><font color="Green" title="E58">P11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:56_1_1_3_2_1"/><i><font color="Green" title="E60">P131</font></i>: 
<font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ;<br/><a NAME="E2:56_1_1_3_2_1"/>
<font color="Maroon" title="c5">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">Y</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:56_1_1_3_2_1"/><i><font color="Green" title="E61">P132</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D6" target="_self" title="RINFSUP2:def.6">Definferior</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:56_1_1_3_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:56_1_1_3_2_1_1"/><i><font color="Green" title="E62">W2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="rinfsup2.html#E4:56_1_1_3_2_1"><i><font color="Green" title="E61">P132</font></i></a></i>;<br/><a NAME="E4:56_1_1_3_2_1_1"/>
<font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E1:56_1_1_3_2_1"><i><font color="Green" title="E60">P131</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_3_2_1_1"><i><font color="Green" title="E62">W2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_3_2_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E5:56_1_1_3_2"><i><font color="Green" title="E59">P12</font></i></a>, <a class="txt" href="rinfsup2.html#E3:56_1_1_3_2_1_1"><i><font color="Green" title="E62">W2</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:56_1_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/><b>hence </b><a NAME="E7:56_1_1_3_2_1"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a>, <a class="txt" href="rinfsup2.html#E4:56_1_1_3_2_1"><i><font color="Green" title="E61">P132</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rinfsup2.html#E3:56_1_1_3_2"><i><font color="Green" title="E58">P11</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:56_1_1_3"/><b>then </b>
 <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E9:56_1_1_3"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a>, <a class="txt" href="rinfsup2.html#E7:56_1_1_3"><i><font color="Green" title="E57">D2a</font></i></a></i>;<br/><b>hence </b><a NAME="E10:56_1_1_3"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T37" target="_self" title="RINFSUP2:th.37">Lm310b</a>, <a class="txt" href="rinfsup2.html#E6:56_1_1_3"><i><font color="Green" title="E56">S1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>assume </b><a NAME="E2:56"/><i><font color="Green" title="E53">R1</font></i>: 
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">a</font> =  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2"/>
(  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1"><b>suppose </b></a><a NAME="E1:56_2_1"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_2_1"/>
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E60"><i><font color="Green" title="E51">Lm330v</font></i></a>, <a class="txt" href="rinfsup2.html#E2:56"><i><font color="Green" title="E53">R1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2"><b>suppose </b></a><a NAME="E1:56_2_2"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_2_2"/>
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E58"><i><font color="Green" title="E49">Lm330u</font></i></a>, <a class="txt" href="rinfsup2.html#E2:56"><i><font color="Green" title="E53">R1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_3"><b>suppose </b></a><a NAME="E1:56_2_3"/>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_2_3"/>
<font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E59"><i><font color="Green" title="E50">Lm330t</font></i></a>, <a class="txt" href="rinfsup2.html#E2:56"><i><font color="Green" title="E53">R1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
