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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">g</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c3">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="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">g</font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b>
<font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">k1</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> <b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E13">A2</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>   st <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="limfunc2.html#T1" title="LIMFUNC2:th.1">LIMFUNC2:1</a></i>;<br/>
<a NAME="E5:11"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">k2</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> <b> such that </b><br/><a NAME="E7:11"/><i><font color="Green" title="E14">A3</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>   st <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="limfunc2.html#T2" title="LIMFUNC2:th.2">LIMFUNC2:2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">k</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c4">k1</font>,<font color="Maroon" title="c5">k2</font>;<br/>

<b>let </b><font color="Maroon" title="c7">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/>

<b>assume </b><a NAME="E8:11"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 ;<br/>

<a NAME="E9:11"/>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:11"/><b>then </b>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E8:11"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:11"/>
<font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:11"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>

<a NAME="E12:11"/>
<font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E13:11"/><b>then </b>
<font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E8:11"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
<font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:11"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>


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