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

<b>assume </b><a NAME="E1:19"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:19"/>
(  <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V1">bounded_above</a> implies <font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T51">SEQ_4:51</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E4:19"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E5:19"/><b>then </b>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:19"/><b>then </b>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/>
<b>hence </b><a NAME="E7:19"/>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>


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