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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>given </b><font color="Maroon" title="c4">m</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> <b> such that </b><a NAME="E2:32"/><i><font color="Green" title="E19">A2</font></i>: 
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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
(  not <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> or <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/>
not <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1"/>
(  not <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> or  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_2_1"/>
 <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
 ;<br/><b>then consider </b><font color="Maroon" title="c5">k</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> <b> such that </b><br/><a NAME="E3:32_1_2_1"/><i><font color="Green" title="E21">A4</font></i>: 
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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)</span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E1:32"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="bhsp_4.html#E1:32_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="bhsp_2.html#T19" title="BHSP_2:th.19">BHSP_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <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> ;<br/><b>assume </b><a NAME="E1:32_1_2_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E2:32_1_2_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:32_1_2_1_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E1:32_1_2_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:32_1_2_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:32_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E1:32_1_2_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:32_1_2_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)</span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E3:32_1_2_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/><a NAME="E7:32_1_2_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/><b>hence </b><a NAME="E8:32_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bhsp_4.html#E2:32"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="bhsp_4.html#E3:32_1_2_1_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:32_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_2"/>
(  not <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> or  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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