<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<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>let </b><font color="Maroon">g</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E42</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:41"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="normsp_1.html#T39">NORMSP_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:41_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:41_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="E4:41_1"/><i><font color="Green">E54</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/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="txt" href="lopban_1.html#E1:41"><i><font color="Green">E33</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">m1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><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="E1:41_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:41_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T13">NORMSP_1:13</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:41_1"/>
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="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div>
