Results 1 to 2 of 2

Thread: Sup(S) proof

  1. #1
    Sep 2009

    Sup(S) proof

    S is a nonempty set in the real numbers. Prove:
    M = sup(S) iff M is an upper bound of S and there exists a sequence (call it a_n) such that lim(a_n) = M as n goes to infinity.
    Having trouble with this one, any suggestions?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Apr 2009
    Here's one direction..

    If M is an upper bound and this sequence exists, then for any epsilon > 0 there exists an N such than for all n > N,

    M - a_n < epsilon

    Assume B is a lesser upper bound than M. Then take epsilon = M - B and we get a_n > B for n large enough. But this contradicts B being an upper bound. Hence M is the least upper bound.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: Oct 19th 2010, 11:50 AM
  2. Replies: 0
    Last Post: Jun 29th 2010, 09:48 AM
  3. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Feb 27th 2010, 11:07 PM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: Jun 8th 2008, 02:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: Apr 14th 2008, 05:07 PM

Search Tags

/mathhelpforum @mathhelpforum