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<b>let </b><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) implies (  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> ) )</span><br/>



<b>assume </b><a NAME="E1:3"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> )</span><br/>

<a NAME="E2:3"/><b>then </b>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> 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">seq1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</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="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</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="c2">seq2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> )
 
<b>by </b><i><a class="ref" href="mesfun10.html#T2" target="_self" title="MESFUN10:th.2">limseq1b</a></i>;<br/>
<b>hence </b><a NAME="E3:3"/>
(  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> )
 <b>by </b><i><a class="ref" href="mesfun10.html#T1" target="_self" title="MESFUN10:th.1">Th61a</a>, <a class="ref" href="mesfunc5.html#T61" title="MESFUNC5:th.61">MESFUNC5:61</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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