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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="rinfsup1.html#NV1" title="RINFSUP1:NV.1">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#K3" title="SERIES_1:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Olive" title="b1">Eseq</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="rinfsup1.html#NV1" title="RINFSUP1:NV.1">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#K3" title="SERIES_1:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c2">Eseq</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( <font color="Maroon" title="c1">seq</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c2">Eseq</font></span><br/>

<a NAME="E3:17"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="rinfsup1.html#D3" title="RINFSUP1:def.3">RINFSUP1:def 3</a></span>;<br/>
<a NAME="E4:17"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<a NAME="E5:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
;<br/>
<a NAME="E6:17"/><span class="kw">then </span>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">upper_bound</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">Eseq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="prob_4.html#T9" target="_self" title="PROB_4:th.9">Th10</a>, <a class="ref" href="prob_4.html#T10" target="_self" title="PROB_4:th.10">Th11</a></span>;<br/>
<a NAME="E7:17"/><span class="kw">then </span>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">upper_bound</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">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#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c2">Eseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D17" title="SUPINF_2:def.17">SUPINF_2:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:17"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c2">Eseq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T24" title="RINFSUP1:th.24">RINFSUP1:24</a>, <a class="ref" href="series_1.html#T16" title="SERIES_1:th.16">SERIES_1:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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