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<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>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:117"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a>
 ;<br/>

<a NAME="E2:117"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:117_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="limfunc1.html#D5" target="_self">Def5</a>, <a class="ref" href="limfunc1.html#T19" target="_self">Th19</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E3:117_1"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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/><a NAME="E1:117_2"/><i><font color="Green">E47</font></i>: 
 <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">f</font>
 ;<br/><b>assume </b><a NAME="E2:117_2"/>
( <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/><a NAME="E3:117_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/><a NAME="E4:117_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</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> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="limfunc1.html#D5" target="_self">Def5</a>, <a class="ref" href="limfunc1.html#T35" target="_self">Th35</a>, <a class="ref" href="limfunc1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E5:117_2"/><b>then </b>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</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="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/><b>hence </b><a NAME="E6:117_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T36" target="_self">Th36</a>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="limfunc1.html#T36" target="_self">Th36</a>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:117"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T34" target="_self">Th34</a>, <a class="ref" href="limfunc1.html#T19" target="_self">Th19</a></i>;<br/>

<b>hence </b><a NAME="E5:117"/>
 <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="limfunc1.html#T27" target="_self">Th27</a></i>;<br/>


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