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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is    <a href="seqm_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font>
 ;<br/>

<b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:70"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D2" target="_self">Def2</a>, <a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">Nseq</font> being  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E6:70"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seqm_3.html#K3" target="_self">*</a> <font color="Maroon">Nseq</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D3" target="_self">Def3</a>, <a class="txt" href="seqm_3.html#E6:70"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">r</font> = <font color="Maroon">r1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E7:70"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:70"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D4" target="_self">Def4</a></i>;<br/>


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