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<a NAME="E1:3_1_1"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a>  is <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c1">rseq</font> =  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a>  as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E2:3_1_1_1"/>
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">rseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D6" title="RSSPACE:def.6">RSSPACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_1_1"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a>  is <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T3" title="COMSEQ_3:th.3">COMSEQ_3:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_1"/>
not  <a href="rsspace3.html#K1" title="RSSPACE3:func.1">the_set_of_l1RealSequences</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="rsspace3.html#D1" target="_self" title="RSSPACE3:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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