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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:33"/>
<font color="Maroon" title="c2">S</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c3">x</font>
 <b>and </b><br/><a NAME="E4:33"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="metric_6.html#T28" target="_self" title="METRIC_6:th.28">Th28</a></i>;<br/>
<a NAME="E5:33"/>
(  <a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:33"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="metric_6.html#T29" target="_self" title="METRIC_6:th.29">Th29</a></i>;<br/>
<a NAME="E6:33"/><b>then </b>
 <a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T27" title="SEQ_2:th.27">SEQ_2:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:33"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>and </b><br/><a NAME="E9:33"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E11:33"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:33_1"/><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="metric_6.html#D14" target="_self" title="METRIC_6:def.14">Def14</a></i>;<br/>
<a NAME="E2:33_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E3:33_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K2" title="METRIC_6:func.2">dist_to_point</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E9:33"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="metric_6.html#E1:33_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:33_1"/>
<font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:33"/>
<font color="Maroon" title="c2">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a>
 <b>by </b><i><a class="txt" href="metric_6.html#E8:33"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="metric_6.html#T20" target="_self" title="METRIC_6:th.20">Th20</a></i>;<br/>


</div>
