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

<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>thus </b><a NAME="E1:82"/>
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 implies <font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<div><i><font color="Green" title="E49">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:82_1_1"/><i><font color="Green" title="E50">A3</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>;<br/><b>thus </b><a NAME="E3:82_1_1"/>
( <font color="Maroon" title="c6">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_1_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:82_1_2"/>
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:82_1_2"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><a NAME="E3:82_1_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><a NAME="E4:82_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="limfunc1.html#T40" target="_self" title="LIMFUNC1:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E5:82_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_1_2"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_1"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:82"/>
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 implies <font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 ;<br/>

<div><i><font color="Green" title="E49">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:82_2_1"/><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>;<br/><b>thus </b><a NAME="E3:82_2_1"/>
( <font color="Maroon" title="c6">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_2_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:82_2_2"/>
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:82_2_2"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><a NAME="E3:82_2_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><a NAME="E4:82_2_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="limfunc1.html#T40" target="_self" title="LIMFUNC1:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E5:82_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_2_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_2"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:82"/>
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 implies <font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_3"/><i><font color="Green" title="E48">A9</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<div><i><font color="Green" title="E49">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:82_3_1"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_3"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>;<br/><b>thus </b><a NAME="E3:82_3_1"/>
( <font color="Maroon" title="c6">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_3_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:82_3_2"/>
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:82_3_2"/><b>then </b><i><font color="Green" title="E50">A12</font></i>: 
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><a NAME="E3:82_3_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_3"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><a NAME="E4:82_3_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:82_3"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="limfunc1.html#T41" target="_self" title="LIMFUNC1:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E5:82_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_3_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_3"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_3"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:82"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 ;<br/>

<div><i><font color="Green" title="E49">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:82_4"/><i><font color="Green" title="E50">A15</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E4:82"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>;<br/><b>thus </b><a NAME="E3:82_4"/>
( <font color="Maroon" title="c6">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_4"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:82_5"/>
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:82_5"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><a NAME="E3:82_5"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E4:82"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><a NAME="E4:82_5"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E4:82"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="limfunc1.html#T41" target="_self" title="LIMFUNC1:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E5:82_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:82_5"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:82"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E5:82"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/>


</div>
