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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:96"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E2:96"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font> implies 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/>( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</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">s</font> ) &amp;  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">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self">Th30</a>, , </i>;<br/>

<b>assume </b><a NAME="E3:96"/><i><font color="Green">E39</font></i>: 
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/>( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</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">s</font> ) &amp;  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">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> )
 ;<br/>

<a NAME="E4:96"/><b>then </b>
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/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</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">s</font>
 
;<br/>
<a NAME="E5:96"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self">Th30</a>, </i>;<br/>
<a NAME="E6:96"/>
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">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:96"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</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#T30" target="_self">Th30</a>, </i>;<br/>
<b>hence </b><a NAME="E8:96"/>
<font color="Maroon">r</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="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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