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<b>let </b><font color="Maroon">K</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">L</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">K</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">rseq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E5:18"/><i><font color="Green">E53</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">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E6:18"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , , <a class="ref" href="series_1.html#T1">SERIES_1:1</a></i>;<br/>
<a NAME="E7:18"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">L</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">rseq</font></span>)</span> = 
<font color="Maroon">L</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
0
<b>by </b><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="E10:18"/><i><font color="Green">E57</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">n</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"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E11:18"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:18"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<a NAME="E13:18"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:18"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K3">to_power</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


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