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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a> )
 ;<br/>

<a NAME="E2:18"/>
 <a href="series_1.html#K1">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="prob_4.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E3:18"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></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/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></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/>
<a NAME="E5:18"/>
 <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="prob_4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:18"/><b>then </b>
 <a href="rinfsup1.html#K1">sup</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_4.html#T11" target="_self">Th11</a>, <a class="txt" href="prob_4.html#E4:18"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="rinfsup1.html#K1">sup</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>
<a NAME="E8:18"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rinfsup1.html#D3">RINFSUP1:def 3</a></i>;<br/>
<a NAME="E9:18"/><b>then </b>
 <a href="series_1.html#K1">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="ref" href="series_1.html#T19">SERIES_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:18"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="prob_4.html#E7:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="prob_4.html#E4:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rinfsup1.html#T24">RINFSUP1:24</a></i>;<br/>


</div>
