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

<b>assume </b><a NAME="E1:58"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V5">monotone</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1"/>
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V3">non-decreasing</a> or <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a> )
 </span><b>by </b><i><a class="txt" href="limfunc1.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 ;<br/><div class="add"><a NAME="E2:58_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1_1_1_1"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> or  not <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_1_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/><div class="add"><a NAME="E2:58_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E4"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/><a NAME="E3:58_1_1_1_1_1_1"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_1_1_1_1"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E5"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="seq_4.html#T53">SEQ_4:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_1_2"/>
not <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_1_2"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="limfunc1.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_1_1"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a>
 ;<br/><div class="add"><a NAME="E2:58_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1_2_1_1"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> or  not <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_2_1_1_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 ;<br/><div class="add"><a NAME="E2:58_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E4"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/><a NAME="E3:58_1_1_2_1_1_1"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2_1_1_1"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E5"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="seq_4.html#T53">SEQ_4:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_2_1_1_2"/>
not <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_2"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="limfunc1.html#E2"><i><font color="Green">Lemma33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_1_2"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:58"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> )
 ;<br/>


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