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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clopban1.html#NM2" title="CLOPBAN1:NM.2">ComplexBanachSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:47"/>
<font color="Maroon" title="c2">seq</font> is <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>
 ;<br/>

<a NAME="E2:47"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D4" target="_self" title="CLOPBAN3:def.4">Def4</a></i>;<br/>
<a NAME="E3:47"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:47_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E39">A2</font></i>: 
for <font color="Olive" title="b1">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 <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="clopban3.html#E3:47"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="seq_4.html#T58" title="SEQ_4:th.58">SEQ_4:58</a></i>;<br/><b>take </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font>;<br/><b>let </b><font color="Maroon" title="c6">m</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="E4:47_1"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E5:47_1"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="clopban3.html#E3:47_1"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/><a NAME="E6:47_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban3.html#E4:47_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="clopban3.html#T29" target="_self" title="CLOPBAN3:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E7:47_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="clopban3.html#E5:47_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon" title="c2">seq</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="clopban3.html#T28" target="_self" title="CLOPBAN3:th.28">Th28</a></i>;<br/>


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