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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:19"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tbsp_1.html#V2">convergent</a> &amp;  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tbsp_1.html#V2">convergent</a> &amp;  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tbsp_1.html#D4">TBSP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="metric_6.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>2</sub></font> iff ( <font color="Maroon">c<sub>3</sub></font> is <a href="tbsp_1.html#V2">convergent</a> &amp;  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="metric_6.html#T21" target="_self">Th21</a>, <a class="ref" href="metric_6.html#T26" target="_self">Th26</a></i>;<br/>


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