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Homework Help: How to prove a number is a supremum of a set

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that if
    (i) [tex]\forall n[/tex][tex]\in N[/tex], u - (1/n) is not an upper bound of s
    (ii) [tex]\forall n[/tex][tex]\in N[/tex], u + (1/n) is an upper bound of S
    then, u = supS

    2. Relevant equations

    3. The attempt at a solution
    It (i) and (ii) are true, then
    [tex]\exists s[/tex][tex]\in S[/tex] s.t. u - (1/n) < s
    and u+(1/n)>s for all s.
    I'm not sure where to go from here.
  2. jcsd
  3. Oct 4, 2009 #2


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    Science Advisor
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    Gold Member

    Re: suprenums

    How about a proof by contradiction?

    Suppose [itex]u > \sup S[/itex]. Then there exists [itex]n \in \mathbb{N}[/itex] such that [itex]\sup S < u - 1/n < u[/itex]. Does this violate one of (i) or (ii)?

    Next suppose [itex]u < \sup S[/itex]. Can you rule this out as well?
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