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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="power.html#K2">-root</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green">E21</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/>

<a NAME="E4:35"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:35_1"/>
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_1"/>
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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="E2:35_1_1"/>
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <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/>


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

<b>thus </b><a NAME="E2:35_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K2">-root</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E1:35"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:35_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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="E5:35"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></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="txt" href="bhsp_4.html#E2:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="bhsp_4.html#E3:35"><i><font color="Green">E21</font></i></a>, <a class="ref" href="series_1.html#T32">SERIES_1:32</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_4.html#V2" target="_self">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="bhsp_4.html#D8" target="_self">Def8</a></i>;<br/>


</div>
