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Thread: 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.
    $\displaystyle S\leq b$ $\displaystyle \forall$ $\displaystyle s\in S.$

    Assume $\displaystyle b \in S.$

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

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

    Therefore $\displaystyle b \in S \Rightarrow b= m =\sup{S}$
    Last edited by Anonymous1; Apr 4th 2010 at 10:37 PM.
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    very helpful. clean/crisp explantation. thank you.
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