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Math Help - Supremums: sup AC = supA supC

  1. #1
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    Supremums: sup AC = supA supC

    If A and C are subsets of R, let AC={ac: a E A, c E C}. If A and C are bounded and the sets consist of strictly positive elements, prove that sup AC = supA supC.

    Attempt:
    a(sup A) and c(sup C)
    => ac supA supC for all (ac) E (AC)
    Hence supA supC is an upper bound for the set AC.

    But how to prove that sup AC = supA supC?

    Any help is appreciated!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post
    If A and C are subsets of R, let AC={ac: a E A, c E C}. If A and C are bounded and the sets consist of strictly positive elements, prove that sup AC = supA supC.

    Attempt:
    a(sup A) and c(sup C)
    => ac supA supC for all (ac) E (AC)
    Hence supA supC is an upper bound for the set AC.

    But how to prove that sup AC = supA supC?

    Any help is appreciated!
    Let a\in A and c\in C then a\leqslant \sup\text{ }A and c\leqslant \sup\text{ }C and so ac\leqslant\sup\text{ }A\cdot\sup\text{ }C it follows that \sup\text{ }AC\leqslant\sup\text{ }A\cdot\sup\text{ }C.

    That finishes up the rest of your initial thought, care to try the second part?
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