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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:99"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E2:99"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:99"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rinfsup1.html#T86" target="_self">Th86</a>, <a class="ref" href="rinfsup1.html#T87" target="_self">Th87</a></i>;<br/>

<b>assume </b><a NAME="E3:99"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E4:99"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E5:99"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:99"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rinfsup1.html#T86" target="_self">Th86</a></i>;<br/>
<a NAME="E6:99"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E3:99"><i><font color="Green">E82</font></i></a></i>;<br/>
<a NAME="E7:99"/><b>then </b>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:99"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rinfsup1.html#T87" target="_self">Th87</a></i>;<br/>
<b>hence </b><a NAME="E8:99"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:99"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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