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<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="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<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> ;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E15">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> )
 ;<br/>

<a NAME="E2:18"/>
 <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="prob_4.html#E1:18"><i><font color="Green" title="E15">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="E3:18"/><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="E4:18"/><b>then </b><i><font color="Green" title="E16">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>
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">Eseq</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="prob_4.html#T10" target="_self" title="PROB_4:th.10">Th10</a></i>;<br/>
<a NAME="E6:18"/><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="supinf_1.html#K6" title="SUPINF_1:func.6">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#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">Eseq</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="prob_4.html#T11" target="_self" title="PROB_4:th.11">Th11</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green" title="E17">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#K21" title="SUPINF_2:func.21">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="E8:18"/>
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="prob_4.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rinfsup1.html#D3" title="RINFSUP1:def.3">RINFSUP1:def 3</a></i>;<br/>
<a NAME="E9:18"/><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="seqm_3.html#V3" title="SEQM_3:attr.3">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="E10:18"/>
 <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#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c2">Eseq</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="prob_4.html#E7:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T24" title="RINFSUP1:th.24">RINFSUP1:24</a></i>;<br/>


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