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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">seq1</font>, <font color="Maroon" title="c3">seq2</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><b>that </b><br/><a NAME="E1:20"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c3">seq2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E2:20_1"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">m1</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="E4:20_1"/><i><font color="Green" title="E10">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">m1</font> holds <br/> <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="bhsp_2.html#D2" title="BHSP_2:def.2">BHSP_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">m2</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="E6:20_1"/><i><font color="Green" title="E11">A5</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="c6">m2</font> holds <br/> <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_3.html#E2:20"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="bhsp_3.html#E2:20_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="bhsp_3.html#D2" target="_self" title="BHSP_3:def.2">Def2</a></i>;<br/><b>take </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m2</font>;<br/><b>let </b><font color="Maroon" title="c8">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="E7:20_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><a NAME="E8:20_1"/>
<font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:20_1"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E7:20_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:20_1"/><b>then </b><i><font color="Green" title="E13">A7</font></i>: 
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_3.html#E4:20_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/><a NAME="E11:20_1"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E12:20_1"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m2</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E7:20_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:20_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_3.html#E6:20_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/><a NAME="E14:20_1"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E10:20_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E15:20_1"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T42" title="BHSP_1:th.42">BHSP_1:42</a></i>;<br/><b>hence </b><a NAME="E16:20_1"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E14:20_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
<font color="Maroon" title="c3">seq2</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D1" title="BHSP_2:def.1">BHSP_2:def 1</a></i>;<br/>


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