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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<a NAME="E1:87"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E14"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<a NAME="E2:87"/><b>then </b>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V1">bounded_above</a> iff <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<b>hence </b><a NAME="E3:87"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V1">bounded_above</a> iff <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_3.html#V2" target="_self">absolutely_summable</a> )
 <b>by </b><i><a class="ref" href="comseq_3.html#D11" target="_self">Def11</a></i>;<br/>


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