<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a>
 <b>and </b><br/><a NAME="E2:45_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"/><i><font color="Green" title="E30">A4</font></i>: 
( <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> )
 
<b>by </b><i><a class="txt" href="#E2:45_1_1"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<a NAME="E4:45_1_1"/>
( <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> )
 
<b>by </b><i><a class="txt" href="#E2:45_1_1"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<b>hence </b><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>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E3:45_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="seqm_3.html#D7" title="SEQM_3:def.7">SEQM_3:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
