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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</font> <a href="seqfunc.html#R1">common_on_dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a> <a href="xxreal_0.html#V2">positive</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a> <a href="xxreal_0.html#V2">positive</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:14"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">Z</font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="power.html#K3">to_power</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>

<a NAME="E4:14"/>
<font color="Maroon">F</font> <a href="seqfunc.html#R3">is_unif_conv_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:14"/><b>then </b>
<font color="Maroon">F</font> <a href="seqfunc.html#R2">is_point_conv_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#T23">SEQFUNC:23</a></i>;<br/>
<a NAME="E6:14"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, <a class="ref" href="seqfunc.html#D10">SEQFUNC:def 10</a>, <a class="ref" href="seqfunc.html#D14">SEQFUNC:def 14</a></i>;<br/>
<a NAME="E7:14"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:14"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="power.html#K3">to_power</a> 0 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">Z</font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:14"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:14"/>
( <span class="p1">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seqfunc.html#K11">lim</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
