<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>thus </b><a NAME="E1:100"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font> implies <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font> )
 ;<br/>

<a NAME="E2:100_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E3:100_1"/><i><font color="Green">E40</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:100_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:100_1"/><i><font color="Green">E49</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E6:100_1"/><i><font color="Green">E50</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seqm_3.html#V3">non-decreasing</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> =  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E8:100_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:100_1_1"/><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:100_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, </i>;<br/>
<b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:100_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:100_1_1"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E8:100_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k2</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="seqm_3.html#T12">SEQM_3:12</a>, <a class="ref" href="seqm_3.html#T14">SEQM_3:14</a></i>;<br/>
<a NAME="E9:100_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:100_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>consider </b><font color="Maroon">k3</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:100_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k3</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E4:100_1_1_1"/>
<font color="Maroon">k3</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:100_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a>, <a class="ref" href="rinfsup1.html#T5" target="_self">Th5</a>, <a class="ref" href="rinfsup1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:100_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k2</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:100_1_1_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, <a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:100_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:100_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:100_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>


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

<b>assume </b><a NAME="E2:100"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E3:100"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/>
<a NAME="E4:100"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:100"/><i><font color="Green">E103</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E6:100"/><b>then </b><i><font color="Green">E104</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E7:100"/><i><font color="Green">E146</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:100"/><i><font color="Green">E147</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E10:100"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:100_2"/><i><font color="Green">E149</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:100_2"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E4:100_2"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:100_2_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:100_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a>, <a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:100_2_1"/>
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:100_2"/><b>then </b>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font>
 
;<br/>
<a NAME="E6:100_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E7:100_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E8:100_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, </i>;<br/>
<b>hence </b><a NAME="E9:100_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:100"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/>( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:100_3"/><i><font color="Green">E155</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:100_3"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E4:100_3"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, <a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, , </i>;<br/>
<b>hence </b><a NAME="E5:100_3"/>
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E13:100"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:100_4"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:100_4"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:100_4"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:100"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T43">XREAL_1:43</a></i>;<br/>
<a NAME="E15:100"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">p</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E16:100"/><b>then </b><i><font color="Green">E158</font></i>: 
 <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T43">XREAL_1:43</a></i>;<br/>
<a NAME="E17:100"/>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E18:100"/><b>then </b>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:100"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E20:100"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/>


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