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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> , <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>means </b>( <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $2 &amp; $2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x0</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> );<br/>
<div><i><font color="Green" title="E23">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <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> ;<br/><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/><a NAME="E2:33_1_1_1"/>
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:33_1_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:33_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="limfunc3.html#D1" target="_self" title="LIMFUNC3:def.1">Def1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">g1</font> = <font color="Maroon" title="c4">g1</font> as  <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>take </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c6">g1</font>;<br/><b>thus </b><a NAME="E6:33_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c7">g1</font>]
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:33_1_1_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">s</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1" title="RCOMP_1:sch.1">RCOMP_1:sch 1</a>(<a class="txt" href="limfunc3.html#E1:33_1_1"><i><font color="Green" title="E23">A2</font></i></a>);<br/></i>
<a NAME="E4:33_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:33_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="limfunc3.html#T6" target="_self" title="LIMFUNC3:th.6">Th6</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
for <font color="Olive" title="b1">seq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> )
 <b>and </b><br/><a NAME="E6:33_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
for <font color="Olive" title="b1">seq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> )
 ;<br/>

<a NAME="E7:33_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E4:33_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="limfunc3.html#E5:33_1_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:33_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="limfunc3.html#E6:33_1_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>


</div>
