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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E46</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="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E3:40"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:40_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">k1</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:40_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E6:40_1"/><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:40_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:40_1"/><b>then </b>
1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:40_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:40_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E11:40_1"/><b>then </b><i><font color="Green">E53</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E12:40_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/><a NAME="E13:40_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E14:40_1"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:40_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/><a NAME="E17:40_1"/><b>then </b><i><font color="Green">E56</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:40_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:40_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E20:40_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E21:40_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E22:40_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><b>hence </b><a NAME="E23:40_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a>, <a class="txt" href="seq_4.html#E1:40"><i><font color="Green">E45</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


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