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Math Help - working with upper bounds

  1. #1
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    working with upper bounds

    Suppose that b is an upper bound for a set S of real numbers. Prove that if b E S, then b = lub S.
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    Quote Originally Posted by WartonMorton View Post
    Suppose that b is an upper bound for a set S of real numbers. Prove that if b E S, then b = lub S.
    S\leq b \forall s\in S.

    Assume b \in S.

    Set m=\sup{S} so, S\leq b \leq m by assumption.

    Then, since b is an upper bound, and m is the l.u.b. S\leq m \leq b.

    Therefore b \in S \Rightarrow b= m =\sup{S}
    Last edited by Anonymous1; April 4th 2010 at 11:37 PM.
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    very helpful. clean/crisp explantation. thank you.
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