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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:39"/><i><font color="Green">E44</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i/>;<br/>
<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> ;<br/>

<a NAME="E2:39"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/>
<a NAME="E3:39"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_2.html#D3">BHSP_2:def 3</a></i>;<br/>
<a NAME="E5:39"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:39"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="bhsp_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/>


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