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<b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">g</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><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/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E50</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E51</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/>

<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">M</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green">E59</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>set </b><font color="Maroon">N1</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,<font color="Maroon">M</font>;<br/><a NAME="E4:57_1"/><i><font color="Green">E60</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,<font color="Maroon">M</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,<font color="Maroon">M</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>take </b><font color="Maroon">N1</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,<font color="Maroon">M</font>;<br/><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/><b>assume </b><a NAME="E5:57_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font>
 ;<br/><a NAME="E6:57_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:57_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:57_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon">g</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E6:57"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


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