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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:54"/>
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:54"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq</font> is  <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_3.html#D6" target="_self" title="BHSP_3:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54"/>
<font color="Maroon" title="c2">seq</font> is  <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_3.html#T24" target="_self" title="BHSP_3:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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