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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">a</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a><br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E38</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r</font> &amp; <font color="Maroon">a</font> is <a href="seq_2.html#V4">convergent</a> )
 ;<br/>

<a NAME="E2:6"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="seq_2.html#K2">lim</a> <font color="Maroon">b</font> = 
<font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:6_2"/>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:6_2"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E3:6_2"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/><a NAME="E4:6_2"/><b>then </b>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r1</font> )
 ;<br/><b>hence </b><a NAME="E5:6_2"/>
<font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:6"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g1</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E10:6"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/>


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