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

<b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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>assume </b><a NAME="E1:108"/><i><font color="Green" title="E61">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> &amp; <font color="Maroon" title="c3">f2</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">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="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green" title="E62">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</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="E2:108_1"/><i><font color="Green" title="E63">A3</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:108"><i><font color="Green" title="E61">A1</font></i></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="E3:108_1"/>
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">g</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="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:108_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T40" title="RFUNCT_1:th.40">RFUNCT_1:40</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:108"/><i><font color="Green" title="E63">A4</font></i>: 
(  <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c3">f2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:108"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="limfunc1.html#T116" target="_self" title="LIMFUNC1:th.116">Th116</a></i>;<br/>
<a NAME="E4:108"/><b>then </b>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:108"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="limfunc1.html#E2:108"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="limfunc1.html#T117" target="_self" title="LIMFUNC1:th.117">Th117</a></i>;<br/>
<b>hence </b><a NAME="E5:108"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40" title="RFUNCT_1:th.40">RFUNCT_1:40</a></i>;<br/>

<b>thus </b> <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
 <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T40" title="RFUNCT_1:th.40">RFUNCT_1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E1:108"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="limfunc1.html#E2:108"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="limfunc1.html#E3:108"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="limfunc1.html#T117" target="_self" title="LIMFUNC1:th.117">Th117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c3">f2</font></span>)</span>

;<br/>


</div>
