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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</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><a NAME="E1:57"/>
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seqm_3.html#V4">non-increasing</a> &amp; ( 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> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 ;<br/>

<a NAME="E2:57"/><b>then </b>
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="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seqm_3.html#V4">non-increasing</a> &amp; <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</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> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="lopban_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:57"/><b>then </b>
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="series_1.html#V1">summable</a> iff <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#T35">SERIES_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:57"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="lopban_3.html#V2" target="_self">norm_summable</a> iff <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i><a class="ref" href="lopban_3.html#D4" target="_self">Def4</a></i>;<br/>


</div>
