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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">St</font> being  <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>  st <font color="Olive" title="b1">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">St</font> &amp; <font color="Olive" title="b2">St</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  holds <br/> <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Olive" title="b2">St</font> <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="Olive" title="b1">S</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">St</font> being  <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>  st <font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">St</font> &amp; <font color="Olive" title="b1">St</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  holds <br/> <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Olive" title="b1">St</font> <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">S</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c3">St</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  implies  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c3">St</font> <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">S</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">St</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c3">St</font> <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">S</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">xt</font> =  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">S</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="E4:12"/>
<font color="Maroon" title="c2">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="normsp_2.html#T5" target="_self" title="NORMSP_2:th.5">Th5</a></span>;<br/>
<a NAME="E5:12"/><span class="kw">then </span>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></span>;<br/>
<a NAME="E6:12"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="normsp_2.html#T4" target="_self" title="NORMSP_2:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c3">St</font> <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">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#T26" title="METRIC_6:th.26">METRIC_6:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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