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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">p'</font> = <font color="Maroon" title="c3">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>given </b><font color="Maroon" title="c5">s</font> being   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">n</font><b> such that </b><a NAME="E2:76"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c5">s</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 <b>and </b><br/><a NAME="E3:76"/><i><font color="Green" title="E47">A2</font></i>: 
for <font color="Olive" title="b1">x</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="c5">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">x</font>
 <b>and </b><br/><a NAME="E4:76"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c5">s</font>
 ;<br/>

<a NAME="E5:76"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">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>  st <br/>for <font color="Olive" title="b3">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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b3">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p'</font>,<font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">r'</font> = <font color="Maroon" title="c6">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:76_1"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">l</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="E4:76_1"/><i><font color="Green" title="E49">A4</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r'</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E2:76"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="kurato_2.html#E4:76"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="toprns_1.html#D10" title="TOPRNS_1:def.10">TOPRNS_1:def 10</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">v</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> 0,<font color="Maroon" title="c8">l</font>;<br/>

<b>let </b><font color="Maroon" title="c10">m</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="E5:76_1"/>
<font color="Maroon" title="c9">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">m</font>
 ;<br/>

<a NAME="E6:76_1"/><b>then </b>
<font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>
<a NAME="E7:76_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r'</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E4:76_1"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>
<a NAME="E8:76_1"/><b>then </b><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p'</font>,<font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T33" target="_self" title="KURATO_2:th.33">Th33</a></i>;<br/>
<a NAME="E9:76_1"/>
<font color="Maroon" title="c5">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:76"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:76_1"/>
<font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p'</font>,<font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E8:76_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:76"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T50" target="_self" title="KURATO_2:th.50">Th50</a></i>;<br/>


</div>
