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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq1</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>let </b><font color="Maroon" title="c4">k</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/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c4">k</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">seq</font>
 
<b>by </b><i><a class="ref" href="lopban_3.html#T13" target="_self" title="LOPBAN_3:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E3:26"/>
<font color="Maroon" title="c2">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c4">k</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="lopban_3.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="lopban_3.html#T11" target="_self" title="LOPBAN_3:th.11">Th11</a></i>;<br/>

<b>thus </b><a NAME="E4:26"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="txt" href="lopban_3.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="lopban_3.html#E2:26"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="lopban_3.html#T12" target="_self" title="LOPBAN_3:th.12">Th12</a></i>;<br/>


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