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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="E1:111"/><i><font color="Green" title="E62">A1</font></i>: 
( <font color="Maroon" title="c2">f</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> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/>

<div><i><font color="Green" title="E63">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_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="E2:111_1"/><i><font color="Green" title="E64">A3</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">g</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> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:111"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>;<br/><a NAME="E3:111_1"/>
not <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:111_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:111_1"/><b>then </b>
not <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E5:111_1"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:111_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:111_1"/>
( <font color="Maroon" title="c3">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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:111_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E64">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_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/><b>assume </b><a NAME="E1:111_2"/><i><font color="Green" title="E65">A5</font></i>: 
( <font color="Maroon" title="c3">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">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"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> )
 ;<br/><a NAME="E2:111_2"/><i><font color="Green" title="E66">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/><a NAME="E3:111_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E4:111_2"/><b>then </b>
 <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="txt" href="limfunc1.html#E1:111_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="limfunc1.html#E2:111_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:111_2"/><b>then </b><i><font color="Green" title="E67">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#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:111"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="limfunc1.html#E1:111_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="limfunc1.html#D12" target="_self" title="LIMFUNC1:def.12">Def12</a></i>;<br/><a NAME="E6:111_2"/><i><font color="Green" title="E68">A8</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> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:111_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T26" title="RFUNCT_2:th.26">RFUNCT_2:26</a></i>;<br/><a NAME="E7:111_2"/><b>then </b>
<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="seq_1.html#K18" title="SEQ_1:func.18">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:111"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="limfunc1.html#E5:111_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="seq_2.html#T35" title="SEQ_2:th.35">SEQ_2:35</a></i>;<br/><b>hence </b><a NAME="E8:111_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </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#E1:111_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T27" title="RFUNCT_2:th.27">RFUNCT_2:27</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"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </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"><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> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E1:111_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T27" title="RFUNCT_2:th.27">RFUNCT_2:27</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">f</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:111"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="limfunc1.html#E5:111_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="txt" href="limfunc1.html#E6:111_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="seq_2.html#T36" title="SEQ_2:th.36">SEQ_2:36</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:111"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:111"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="limfunc1.html#D6" target="_self" title="LIMFUNC1:def.6">Def6</a></i>;<br/>

<b>hence </b><a NAME="E5:111"/>
 <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c2">f</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:111"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="limfunc1.html#D12" target="_self" title="LIMFUNC1:def.12">Def12</a></i>;<br/>


</div>
