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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="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:15"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tbsp_1.html#V2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:15"/><i><font color="Green">E12</font></i>: 
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>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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#D3">TBSP_1:def 3</a></i>;<br/>
<a NAME="E4:15"/>
<font color="Maroon">c<sub>2</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="metric_6.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E5:15"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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