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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">S</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  holds <br/><font color="Olive" title="b1">S</font> is  <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:32"/>
<font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:32"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#T13" target="_self" title="METRIC_6:th.13">Th28</a></span>;<br/>
<a NAME="E5:32"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="metric_6.html#T14" target="_self" title="METRIC_6:th.14">Th29</a></span>;<br/>
<a NAME="E6:32"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T13" title="SEQ_2:th.13">SEQ_2:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:32"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E9:32"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 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#K1" title="SEQ_1:func.1">.</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E11:32"/>
 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="tbsp_1.html#K2" title="TBSP_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="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> <a href="tbsp_1.html#K2" title="TBSP_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>)</span><br/>

<a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<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#K1" title="SEQ_1:func.1">.</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="tbsp_1.html#K2" title="TBSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D6" target="_self" title="METRIC_6:def.6">Def14</a></span>;<br/>
<a NAME="E2:32_1"/><span class="kw">then </span>
 <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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></span>;<br/>
<a NAME="E3:32_1"/><span class="kw">then </span>
 <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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E4:32_1"/><span class="kw">then </span>
 <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="tbsp_1.html#K2" title="TBSP_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1"/>
<font color="Maroon" title="c2">S</font> <a href="tbsp_1.html#K2" title="TBSP_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:32"/>
<font color="Maroon" title="c2">S</font> is  <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="metric_6.html#T8" target="_self" title="METRIC_6:th.8">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
