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

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="rinfsup1.html#T29" target="_self">Th29</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_1.html#T9">SETLIM_1:9</a></i>
<br/>.= 
 <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>
<b>hence </b><a NAME="E3:48"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> implies <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


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