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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</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/><b>let </b><font color="Maroon">rseq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green">E31</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">rseq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">rseq1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/>

<a NAME="E2:56"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:56"/>
not <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i>, <a class="ref" href="series_1.html#T34">SERIES_1:34</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lopban_3.html#D4" target="_self">LOPBAN_3:def 4</a><br/>


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