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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">S0</font> = <font color="Maroon" title="c4">S</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr3.html#T8" target="_self" title="TOPMETR3:th.8">Th8</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c5">S0</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topmetr3.html#T6" target="_self" title="TOPMETR3:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon" title="c4">S</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topmetr3.html#T9" target="_self" title="TOPMETR3:th.9">Th9</a></i>;<br/>

<a NAME="E5:9"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c5">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topmetr3.html#E3:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="topmetr3.html#T9" target="_self" title="TOPMETR3:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topmetr3.html#T6" target="_self" title="TOPMETR3:th.6">Th6</a></i>;<br/>


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