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

<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g1</font> ) )
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/>

<a NAME="E3:7"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:7"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g1</font> ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n</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="E4:7_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <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">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:7_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E6:7_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E7:7_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E8:7_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:7_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g2</font> where <font color="Olive">g2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g2</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E10:7_1"/><b>then </b>
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g2</font> )
 ;<br/><a NAME="E11:7_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E12:7_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:7"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#D5">LIMFUNC1:def 5</a></i>;<br/>


</div>
