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Math Help - Proving that a set is unbounded above

  1. #1
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    Proving that a set is unbounded above

    Prove that the set (-2, ) is unbounded above and has no minimum element.

    Help please
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  2. #2
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    Re: Proving that a set is unbounded above

    Quote Originally Posted by AW63 View Post
    Prove that the set (-2, ) is unbounded above and has no minimum element.

    Suppose that \mathcal{S}=(-2,\infty).

    unbounded above. If t\in\mathcal{S} we know that 0<1 so t<t+1.

    has no minimum element.. If t\in\mathcal{S} then -2<t.

    So -2<\frac{t+2}{2}<t. Can t be minimal in \mathcal{S}~?
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  3. #3
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    Re: Proving that a set is unbounded above

    S has no upper bound
    Proof by contradiction: Suppose S has an upperbound, a. Certainly, 0 is in S so we must have 0< a. Then a< a+ 1 so a+ 1 is not in S. But S contains all numbers larger than -2 so that is a contradiction

    S has no smallest member.
    Proof by contradicton: Suppose S has smallest member b. Since b is in S, -2< b. Now, -2< (b- 2)/2< b is larger than -2 so in S but less than b which contradicts the assumption that b is the smallest member of S.
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