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<b>let </b><font color="Maroon">X</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:27"/>
( ( for <font color="Olive">r</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">r</font> holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font> ) implies for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">X</font> holds <br/><font color="Olive">V</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:27"/>
( ( for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">X</font> holds <br/><font color="Olive">V</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font> ) implies for <font color="Olive">r</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">r</font> holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">X</font> holds <br/><font color="Olive">V</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon">S</font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
for <font color="Olive">r</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">r</font> holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


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