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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c1">X</font> is <a href="bhsp_3.html#V3" title="BHSP_3:attr.3">complete</a> &amp; <font color="Olive" title="b1">seq</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">seq</font> is <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a></span><br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="bhsp_3.html#V3" title="BHSP_3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a> implies <font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="bhsp_3.html#V3" title="BHSP_3:attr.3">complete</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a></span><br/>

<a NAME="E3:44"/>
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E2:44"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="bhsp_3.html#D6" target="_self" title="BHSP_3:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a>
 <b>by </b><i><a class="ref" href="bhsp_3.html#T24" target="_self" title="BHSP_3:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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