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<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>let </b><font color="Maroon">Rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>) &amp; <font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">Rseq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E47</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">Rseq</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/>

<a NAME="E4:31"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <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> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<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>thus </b><a NAME="E1:31_1"/>
<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> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1"/>
<font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E2:31_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T32">BHSP_1:32</a></i>;<br/>
<a NAME="E3:31_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/>
<a NAME="E4:31_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:31_1_1"/>
<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> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="bhsp_2.html#D3">BHSP_2:def 3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:31_1"/>
<font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_2.html#D3">BHSP_2:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E2:31_1_2"/>
<font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D3">BHSP_2:def 3</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E5:31"/><b>then </b>
<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>, , <a class="ref" href="series_1.html#T30">SERIES_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">seq</font> is <a href="bhsp_4.html#V2" target="_self">absolutely_summable</a>
 <b>by </b><i/>;<br/>


</div>
