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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>thus </b><a NAME="E1:39"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D5" target="_self">LIMFUNC1:def 5</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="limfunc1.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:39_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E5:39_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E3:39_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:39_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:39_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:39"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D4" target="_self">LIMFUNC1:def 4</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:39_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="limfunc1.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:39_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E5:39_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E3:39_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:39_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:39_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E7:39_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:39_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E8:39_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:39"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E4:39"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:39_3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E3:39_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="limfunc1.html#E3:39"><i><font color="Green">E24</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:39_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:39_3"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_2"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="limfunc1.html#E3:39"><i><font color="Green">E24</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:39_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:39_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E5:39"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T15">SEQM_3:15</a></i>;<br/>


</div>
