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

<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Eseq</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">Eseq</font> &amp; <font color="Maroon" title="c1">seq</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> holds <br/> <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Olive" title="b1">Eseq</font></span><br/><b>let </b><font color="Maroon" title="c2">Eseq</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">Eseq</font> &amp; <font color="Maroon" title="c1">seq</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> implies  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c2">Eseq</font> )</span><br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">Eseq</font> &amp; <font color="Maroon" title="c1">seq</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c2">Eseq</font></span><br/>

<a NAME="E2:17"/>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E3:17"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T27" title="SEQ_2:th.27">SEQ_2:27</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
;<br/>
<a NAME="E5:17"/><b>then </b>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <font color="Maroon" title="c2">Eseq</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="prob_4.html#T10" target="_self" title="PROB_4:th.10">Th10</a>, <a class="ref" href="prob_4.html#T11" target="_self" title="PROB_4:th.11">Th11</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c2">Eseq</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D23" title="SUPINF_2:def.23">SUPINF_2:def 23</a></i>;<br/>
<a NAME="E7:17"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="rinfsup1.html#D3" title="RINFSUP1:def.3">RINFSUP1:def 3</a></i>;<br/>
<a NAME="E8:17"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/>
<b>hence </b><a NAME="E9:17"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c2">Eseq</font>
 <b>by </b><i><a class="txt" href="#E4:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E6:17"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T24" title="RINFSUP1:th.24">RINFSUP1:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
