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<span class="kw">let </span><font color="Maroon" title="c1">e</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font> holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b1">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> holds <br/><font color="Maroon" title="c2">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font> implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> holds <br/><font color="Maroon" title="c2">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c2">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">ee</font> = <font color="Maroon" title="c1">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">cseq</font> =  <a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font>;<br/>
<span class="lab"><font color="Green" title="E36">A4</font></span>:  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
<br/>.= 
<font color="Maroon" title="c1">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E37">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">ii</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E2:45_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">ii</font></span>)</span> &amp; <font color="Maroon" title="c2">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:45"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">seq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="seq_4.html#T20" onmouseover="rs('seq_4/T20')" onmouseout="rh()">SEQ_4:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:45"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="seq_2.html#T18" onmouseover="rs('seq_2/T18')" onmouseout="rh()">SEQ_2:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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