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Math Help - Continuous Function on a Closed Bounded Interval

  1. #1
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    Continuous Function on a Closed Bounded Interval

    Assuming the theorem that a continuous real-valued function on a closed
    bounded interval is bounded and attains its bounds, prove that if f : R → R is
    continuous and f(x) → +∞ as x → ∞ then there exists some x_0 ∈ R such that
    f(x) > f(x_0) for all x ∈ R.

    I'm not even sure where to start with this - doesn't anybody have any hints.

    Thanks.
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  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
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    Hello,

    Let x_1 be a real number such that for all x\leqslant x_1, f(x)\geqslant f(42).
    Let x_2 be a real number such that for all x\geqslant x_2, f(x)\geqslant f(42).

    Can you show that x_0 exists and that x_0\in[x_1,x_2] ?
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