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Math Help - Proof...monotonicity

  1. #1
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    Proof...monotonicity

    Let f:[a,b]--->R be monotone. Prove that f has a limit at BOTH a and b.

    I have no idea what to do. Please help. Thanks
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  2. #2
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    Quote Originally Posted by zhupolongjoe View Post
    Let f:[a,b]--->R be monotone. Prove that f has a limit at BOTH a and b.
    Let's assume that f is nondecreasing.
    The set S=\{f(x):x\in (a,b)\} is bounded below by f(a).
    So let L=\inf(S). Show that L is limit as x\to a^+.
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  3. #3
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    Ok, so do I just show that using the limit definition or is there some other trick?
    After that, I assume I should make T={f(x): x is in (a,b)} bounded above by f(b)

    And then have L2=sup(T) and show L2 is the limit as x--->b-?

    Thanks
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