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<b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>

<a NAME="E1:88"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E14"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>
<a NAME="E2:88"/><b>then </b>
(  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> iff <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></i>;<br/>
<b>hence </b><a NAME="E3:88"/>
(  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> iff <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a> )
 <b>by </b><i><a class="ref" href="comseq_3.html#D11" target="_self" title="COMSEQ_3:def.11">Def11</a></i>;<br/>


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