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<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">s2</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">E31</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s2</font> is <a href="series_1.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E3:22"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon">s1</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i>, , </i>;<br/>

<a NAME="E5:22"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:22"/><i><font color="Green">E34</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:22"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</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">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<b>by </b><i>, </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">s1</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">s2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<b>hence </b><a NAME="E9:22"/>
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">s2</font>
 ;<br/>


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