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<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>assume </b><a NAME="E1:119"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>
 ;<br/>

<div><i><font color="Green" title="E67">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>now </b></a><div class="add"><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>consider </b><font color="Maroon" title="c4">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:119_1"/><i><font color="Green" title="E68">A3</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c4">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#E1:119"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="limfunc1.html#D9" target="_self" title="LIMFUNC1:def.9">Def9</a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>;<br/><b>thus </b><a NAME="E3:119_1"/>
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:119_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E68">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><a NAME="E1:119_2"/><i><font color="Green" title="E69">A5</font></i>: 
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</font>
 ;<br/><b>assume </b><a NAME="E2:119_2"/>
( <font color="Maroon" title="c3">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="c3">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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E3:119_2"/><b>then </b><i><font color="Green" title="E70">A6</font></i>: 
( <font color="Maroon" title="c3">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="c3">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#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><a NAME="E4:119_2"/><b>then </b><i><font color="Green" title="E71">A7</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</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="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:119"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="limfunc1.html#E1:119_2"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></i>;<br/><a NAME="E5:119_2"/><b>then </b>
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</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="c3">seq</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T26" title="SEQ_4:th.26">SEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E6:119_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</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="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:119_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</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="c3">seq</font></span>)</span> = 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p2">(<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="c3">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E3:119_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E4:119_2"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="seq_4.html#T27" title="SEQ_4:th.27">SEQ_4:27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:119"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:119"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="limfunc1.html#D9" target="_self" title="LIMFUNC1:def.9">Def9</a></i>;<br/>

<b>hence </b><a NAME="E5:119"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:119"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></i>;<br/>


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