<?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">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><a NAME="E1:24"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font> or <font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font> or <font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:24_1_1_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">g2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:24_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>;<br/><b>take </b><font color="Maroon">g2</font> = <font color="Maroon">g2</font>;<br/><b>thus </b><a NAME="E4:24_1_1_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:24_1_1_1_2"/><i><font color="Green">E46</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="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:24_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:24_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:24_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="seq_1.html#K22">abs</a> <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> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T52">LIMFUNC1:52</a></i>;<br/><a NAME="E5:24_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:24_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T25">RFUNCT_2:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1_1"/>
 <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:24_1_1_2_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">g2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:24_1_1_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>;<br/><b>take </b><font color="Maroon">g2</font> = <font color="Maroon">g2</font>;<br/><b>thus </b><a NAME="E4:24_1_1_2_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:24_1_1_2_2"/><i><font color="Green">E65</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="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:24_1_1_2_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:24_1_1_2_2"/><b>then </b>
<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>, , </i>;<br/><a NAME="E4:24_1_1_2_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="seq_1.html#K22">abs</a> <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> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T52">LIMFUNC1:52</a></i>;<br/><a NAME="E5:24_1_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:24_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T25">RFUNCT_2:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1_2"/>
 <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
 <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="limfunc3.html#R2" target="_self">is_divergent_to+infty_in</a> <font color="Maroon">x0</font>
 ;<br/>


</div>
