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Thread: question on a proof

  1. #1
    Sep 2009

    question on a proof

    I need a little confirmation on the following proof:
    The problem is this:
    For an interval I, with f:I -> R where f is strictly increasing with f(I) an interval, prove that f^(-1):R -> I is strictly increasing.

    My proof is this:
    Because f is strictly increasing, for y,z exists in I such that y<z,
    Let y,z exist in I. Then f(y),f(z) exist in R. and f(y) < f(z).
    but f^(-1)[f(y)]<f^(-1)[f(z)] which implies y<z and thus f^(-1) is strictly increasing.
    This proof seems trivial to me, which makes me think I did something wrong. I possibly assumed too much in my proof, not sure. Any ideas?
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  2. #2
    Junior Member
    Oct 2009

    I suffer from similar habbits. Always assuming things are complicated. Your proof is correct in its simplicity. You are absolutely right to assume the existence of $\displaystyle f^{-1}$, its given in the problem statement. Hence you may use the identity function $\displaystyle f^{-1}(f(x)) = x$, by the definition of the inverse.

    In fact just to reassure you that your proof makes sense, remember that f is the inverse of the function $\displaystyle f^{-1}$. And a function only has an inverse if it is strictly increasing or decreasing (a standard theorem from set theory). You have of course proved $\displaystyle f^{-1} $ is strictly increasing. And thus the inverse of $\displaystyle f^{-1}$ exists. namely, f.
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