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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<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>assume </b><a NAME="E1:19"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">seq</font> is <a href="bhsp_2.html#V1" target="_self">convergent</a>
 ;<br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">seq</font> <a href="bhsp_1.html#K7">+</a> <font color="Maroon">x</font> is <a href="bhsp_2.html#V1" target="_self">convergent</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E43</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">m1</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:19_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E5:19_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T49">BHSP_1:49</a></i>
<br/>.= 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
<br/>.= 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
;<br/><b>hence </b><a NAME="E7:19_1"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="bhsp_1.html#K7">+</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="bhsp_1.html#D12">BHSP_1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
 <a href="bhsp_2.html#K1" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="bhsp_1.html#K7">+</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


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