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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> 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/>
<a NAME="E3:6"/><i><font color="Green">E33</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">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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/>
<a NAME="E4:6"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</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/>
<a NAME="E5:6"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<a NAME="E6:6"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:6"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<a NAME="E8:6"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>then 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="E11:6"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</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">k</font> holds <br/>0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 <b>by </b><i>, <a class="ref" href="limfunc1.html#T29">LIMFUNC1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E12:6"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E13:6"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:6"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>;<br/>
<a NAME="E15:6"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:6"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E17:6"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


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