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<b>let </b><font color="Maroon">a</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">b</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) ) )
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:2_1"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>then consider </b><font color="Maroon">k1</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="E3:2_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>consider </b><font color="Maroon">k2</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="E5:2_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E6:2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E7:2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/><b>consider </b><font color="Maroon">n</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="E9:2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="scheme1.html#T1">SCHEME1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:2_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k2</font>
 ;<br/><a NAME="E2:2_1_1_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:2_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:2_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2"/>
<font color="Maroon">d</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="E4:2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</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#D7">SEQ_2:def 7</a></i>;<br/>


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