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Math Help - More proofs

  1. #1
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    More proofs

    1. If E has a supremum, prove that it has only one supremum.

    'Was thinking about taking the approach: Suppose E has 2 supremums and prove that it is false. Help on ideas to approach and make this flow will help me lots.

    2. Prove that if M is an upperbound of a set E and M is in E, then M is the supremum of E.

    Again help on where to go and flow with this one too. I so wish I could wrap my head around these proofs and understand them more. I think having taken all the calc's 15 years ago is part of my problem.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Caity View Post
    1. If E has a supremum, prove that it has only one supremum.
    Suppose E has more than one supremum and let s_1 and s_2 be suprema for E with s_1 \ne s_2.

    Now s_1 is a least upper bound for E, and as s_2 is an upper bound s_1 \le s_2.

    Now the roles of s_1 and s_2 in the last paragraph may be reversed and so we conclude that s_1 \ge s_2 . Hence s_1=s_2 a contradiction, so if E has a supremum it is unique.

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Caity View Post
    2. Prove that if M is an upperbound of a set E and M is in E, then M is the supremum of E.
    Suppose M is not the supremum of E, then there is an upperbound U<M for E.

    But because U is an upperbound for E and M is in E; M \le U a contradiction.

    CB
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