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Math Help - Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

  1. #1
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    Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

    Hi, I have the following problem... I know the second bit of it, but I have no clue how to prove the first bit. I know B is bounded below (it is basically set A but with opposite signs, so the supremum of A will be the lowest number of set B, its infimum), but I don't know how to express it mathematically. Thanks for any help you can give me!

    "Let A be a non-empty subset of R that is bounded above. Let

    B = {-x : x ϵ A}.

    Show that B is bounded below. How are sup A (supremum of A) and inf B (infimum) related?"

    I don't know how to prove that B is bounded below, but for the question "How are sup A (supremum of A) and inf B (infimum) related?", I have the following:

    sup A = -inf B.
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  2. #2
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    Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

    Quote Originally Posted by juanma101285 View Post
    "Let A be a non-empty subset of R that is bounded above. Let
    B = {-x : x ϵ A}.
    Show that B is bounded below. How are sup A (supremum of A) and inf B (infimum) related?" sup A = -inf B.
    If \alpha=\sup(A) then if t\in B then -t\in A.
    So -t\le\alpha or -\alpha\le t.
    Thus -\alpha\le\beta=\inf(B)

    Suppose that -\alpha<\beta then \alpha>-\beta so \left( {\exists x \in A} \right)\left[ { - \beta  < x \leqslant \alpha } \right]
    What is wrong with that?
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    Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

    Thanks for the message!

    Ermm... so that would be a contradiction because that would mean x is sup(A) and -x is inf(B)??
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    Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

    Quote Originally Posted by juanma101285 View Post
    Thanks for the message!

    Ermm... so that would be a contradiction because that would mean x is sup(A) and -x is inf(B)??
    Well it means -x\in B and -x<\beta~.
    Can that happen?
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  5. #5
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    Re: Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below

    Let a be an upper bound of A. Let x be any member of B. Then x= -y for some y in A. Then y< a. If follows that -y= x> -a. Thus, -a is a lower bound for B.
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