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Math Help - Continuous function #1

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Continuous function #1

    Prove:

    If f(x) is continius function on [0,\infty) and if lim_{x\to \infty} f(x)=M<\infty, then f(x) is bounded on [0,\infty]

    What I did..

    I looked the next two intervals:

    1. [0,M], f(x) is bounded there.

    2. [M, \infty), and I don't know what to do next here...


    Thanks!
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Prove:

    If f(x) is continius function on [0,\infty) and if lim_{x\to \infty} f(x)=M<\infty, then f(x) is bounded on [0,\infty]

    What I did..

    I looked the next two intervals:

    1. [0,M], f(x) is bounded there.

    2. [M, \infty), and I don't know what to do next here...


    Thanks!
    Use the definition of \lim_{x\to \infty} f(x)=M, taking \varepsilon=1. It tells you that there exists N such that |f(x)-M|<1 whenever x>N. So f is bounded on [N,\infty). But it is also bounded on [0,N] ... .
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