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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22"/><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 <br/>( 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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="series_1.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E3:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:22"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i><a class="txt" href="series_1.html#E1:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="series_1.html#E2:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="series_1.html#T22" target="_self">Th22</a></i>;<br/>

<a NAME="E5:22"/><b>then </b><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="ref" href="series_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E22</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E2:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="series_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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="Olive">b<sub>1</sub></font> <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>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="series_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:22"/><b>then </b>
 <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> <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>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="series_1.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="series_1.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:22"/><b>then </b>
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <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>2</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E10:22"/>
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
