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<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">ap</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:16"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> is <a href="series_1.html#V1">summable</a>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E28</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  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E29</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  <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E5:16"/><i><font color="Green">E30</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E31</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E33</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  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:16"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>thus </b><a NAME="E9:16"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/>

<a NAME="E10:16"/><i><font color="Green">E39</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:16_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E3:16_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E4:16_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/><a NAME="E5:16_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E6:16_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="holder_1.html#E1:16"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:16_1"/>
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:16"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">ap</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/>

<a NAME="E13:16"/>
<font color="Maroon">ap</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="E14:16"/><b>then </b>
<font color="Maroon">ap</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="E15:16"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T54">SEQ_4:54</a></i>;<br/>
<b>hence </b><a NAME="E16:16"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</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#K2">Sum</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


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