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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>and </b><br/><a NAME="E2:20"/><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 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E3:20"/><i><font color="Green">E20</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="series_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E21</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="series_1.html#D2" target="_self">Def2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:20_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="series_1.html#E3:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/><a NAME="E2:20_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E2:20"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E3:20_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:20_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:20_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:20"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="series_1.html#E4:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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