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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:24"/>
<font color="Maroon">seq</font> is <a href="bhsp_3.html#V2" target="_self">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">M</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:24"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E51</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  <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_1.html#K6">-</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> = 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D10">BHSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T37">BHSP_1:37</a></i>
;<br/><b>hence </b><a NAME="E2:24_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_1.html#K6">-</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:24"/>
 <a href="bhsp_1.html#K6">-</a> <font color="Maroon">seq</font> is <a href="bhsp_3.html#V2" target="_self">bounded</a>
 <b>by </b><i>, </i>;<br/>


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