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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">S</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="c3">St</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font>
 ;<br/>

<div><i><font color="Green" title="E11">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c3">St</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">xt</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c3">St</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c4">xt</font>
 <b>by </b><i><a class="ref" href="metric_6.html#T22" title="METRIC_6:th.22">METRIC_6:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">xt</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E5:11_1"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <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>  st <br/>for <font color="Olive" title="b3">n</font> being   <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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="normsp_2.html#T4" target="_self" title="NORMSP_2:th.4">Th4</a>, <a class="txt" href="normsp_2.html#E3:11_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:11_1"/>
<font color="Maroon" title="c2">S</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9" title="NORMSP_1:def.9">NORMSP_1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2"/>
<font color="Maroon" title="c2">S</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:11_2"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <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>  st <br/>for <font color="Olive" title="b3">n</font> being   <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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9" title="NORMSP_1:def.9">NORMSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">xt</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/><a NAME="E5:11_2"/>
<font color="Maroon" title="c3">St</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c5">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="normsp_2.html#T4" target="_self" title="NORMSP_2:th.4">Th4</a>, <a class="txt" href="normsp_2.html#E3:11_2"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:11_2"/>
<font color="Maroon" title="c3">St</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="metric_6.html#T21" title="METRIC_6:th.21">METRIC_6:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
( <font color="Maroon" title="c3">St</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> iff <font color="Maroon" title="c2">S</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>


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