<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="bhsp_3.html#NR5" target="_self">is_Hilbert_space</a> 
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq</font> is <a href="bhsp_4.html#V2" target="_self">absolutely_summable</a>
 ;<br/>

<a NAME="E3:42"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/> <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="series_1.html#T25">SERIES_1:25</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E2:42_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/><a NAME="E3:42_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E4:42_1_1"/>
 <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E5:42_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/><a NAME="E6:42_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:42_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:42_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">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">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:42_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:42_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<font color="Maroon">seq</font> is <a href="bhsp_4.html#V1" target="_self">summable</a>
 <b>by </b><i>, </i>;<br/>


</div>
