Least upper bound problem

  • Thread starter boombaby
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  • #1
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Homework Statement



Find subsets E[tex]\subset[/tex]S1[tex]\subset[/tex]S2[tex]\subset[/tex]S3[tex]\subset[/tex]Q such that E has a least upper bound in S1, but does not have any least upper bound in S2, yet does have a least upper bound in S3.

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The Attempt at a Solution



I got totally stuck with it. If a[tex]\in[/tex]S1 is the least upper bound of E, does that mean a is also in S2 and hence a least upper bound of E in S2?
On the other hand, if E has a least upper bound b in Q, is b unique? So in any subset S of Q, just check if b is in S to see if E has a least upper bound in S?
Thanks a lot
 

Answers and Replies

  • #2
Dick
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I'll get you started. Let E=[0,1) and S1=union(E,{2}). E has a LUB in S1 of 2. Do you see how the LUB can depend on the set E is included in? Can you finish?
 
  • #3
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O I got it now....So S2=union(S1,{x[tex]\in[/tex]Q|x^2>2, 0<=x<=2}) would work. And S3=union(S2,{1.1})
Thanks very much:rofl:
 

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