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Math Help - Equicontinuity

  1. #1
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    Equicontinuity

    Hi, I've been working on:
    Construct a bounded sequence of continuous functions  f_n:[0,1] \rightarrow \mathbb{R} \\ s.t. \left| \left| f_n - f_m \right| \right| = sup \left| f_n(x) - f_m(x) \right| = 1, n \neq m, x \in [0,1]

    Can such a sequence be equicontinuous?

    So far I have the example that  f_n(x) = sin(nx) but I don't know how to handle the equicontinuity. I know this particular family of functions does not have a uniformly convergent subsequence so it can't be equicontinuous, but I don't know how to rigorously prove it. Any help would be much appreciated!
    Last edited by deannalouise; May 28th 2009 at 05:51 PM.
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  2. #2
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    Look at the sequence of functions f_n(x) = sin(nx)

    Let x_n = \frac{3\pi}{2n}, y_n = \frac{\pi}{n}. Then  |x_n - y_n| = \frac{\pi}{2n} which converges to 0 as  n \rightarrow \infty, but |f_n(x_n) - f_n(y_n)| = 1,  \forall n \ge 1

    Hence the sequence is not equicontinuous. Hope this helps!
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  3. #3
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    Yes, thanks so much!
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