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Math Help - Continuity proofs

  1. #1
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    Continuity proofs

    Prove that for continuous f, f(x_n)>=f(y_n) => f(lim x_n)>=f(lim y_n)

    I started a proof by contradiction, because that is how I remember doing these types of problems in the past. I said, suppose not, then there is an N such that for n>N, f(y_n)>f(x), but how can I go from there to showing that means there is an n such that f(x_n)>f(y_n). I have a feeling that it is a clever choice of epsilon that I cannot think of.

    Thanks!
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  2. #2
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    Re: Continuity proofs

    Quote Originally Posted by powerhawk View Post
    [FONT=Consolas]Prove that for continuous f, f(x_n)>=f(y_n) => f(lim x_n)>=f(lim y_n)
    You know that \lim (f({x_n})) = X\;\& \;\lim (f({y_n})) = Y both exist.
    Now suppose that X<Y. Then let \varepsilon  = \frac{{Y - X}}{2}.
    (\exists~N)[n\ge N] implies X - \varepsilon  < {x_n} < X + \varepsilon\;\; \&  Y - \varepsilon  < {y_n} < Y + \varepsilon .

    What is the contradiction?
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  3. #3
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    Re: Continuity proofs

    Got it. Thank you so much!
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