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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">and </span><br/><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/>

<a NAME="E3:45_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
( <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<a NAME="E4:45_1_1"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_1_1"/>
<font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="seqm_3.html#D5" title="SEQM_3:def.5">SEQM_3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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