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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">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<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><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V7</a>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="normsp_1.html#T27" title="NORMSP_1:th.27">NORMSP_1:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:12"/><i><font color="Green" title="E16">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>  holds  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="normsp_1.html#D4" title="NORMSP_1:def.4">NORMSP_1:def 4</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="bhsp_2.html#E4:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bhsp_2.html#T1" target="_self" title="BHSP_2:th.1">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>take </b><font color="Maroon" title="c7">m</font> = 0;<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="E2:12_1"/>
<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="E3:12_1"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E4:12_1"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">seq</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/><a NAME="E5:12_1"/><b>then </b>
<font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:12_1"/><b>then </b>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E6:12"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><a NAME="E7:12_1"/><b>then </b>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E4:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="bhsp_2.html#E7:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:12_1"/><b>then </b>
<font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E6:12"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E9:12_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">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E1:12_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="bhsp_1.html#T41" title="BHSP_1:th.41">BHSP_1:41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:12"/>
 <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" title="c2">x</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E8:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="bhsp_2.html#D2" target="_self" title="BHSP_2:def.2">Def2</a></i>;<br/>


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