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<b>let </b><font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c2">K</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green" title="E85">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E2:90"/><i><font color="Green" title="E86">A2</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green" title="E87">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:90"/><i><font color="Green" title="E88">A4</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ;<br/><a NAME="E2:90_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:90"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E2:90"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/><a NAME="E3:90_1"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:90"><i><font color="Green" title="E87">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:90_1"/>
<font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:90"><i><font color="Green" title="E87">A3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:90"/>
<font color="Maroon" title="c2">K</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:90"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:90"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E4:90"><i><font color="Green" title="E88">A4</font></i></a>, <a class="ref" href="mesfunc5.html#T68" target="_self" title="MESFUNC5:th.68">Th68</a></i>;<br/>

<b>thus </b><a NAME="E7:90"/>
( <font color="Maroon" title="c2">K</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:90"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>

<a NAME="E8:90"/>
(  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:90"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E5:90"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E9:90"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:90"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:90"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E4:90"><i><font color="Green" title="E88">A4</font></i></a>, <a class="ref" href="mesfunc5.html#T68" target="_self" title="MESFUNC5:th.68">Th68</a></i>;<br/>


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