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

<b>set </b><font color="Maroon">f</font> = <a href="numbers.html#K5">NAT</a>  <a href="borsuk_1.html#K1">--&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <a href="numbers.html#K5">NAT</a>  <a href="borsuk_1.html#K1">--&gt;</a> 0 as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E2:7_1_1"/>
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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon">f</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="rsspace.html#T20">RSSPACE:20</a></i>;<br/>

<b>thus </b><a NAME="E4:7_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V3">constant</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">r</font> = 0; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a><br/>

<b>thus </b><a NAME="E1:7_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E5:7_1_1"/>
<font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>


</div>
