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Math Help - need help with two (basic?) proofs involving equicontinuity

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    Newbie sonictech's Avatar
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    need help with two (basic?) proofs involving equicontinuity

    I'm stuck trying to wrap my head around two proofs involving equicontinuity.

    1) If an equicontinuous sequence of functions (fn) converges pointwise to f on a set S, then f is uniformly continuous on S.

    2) If a sequence of continuous functions (fn) converges uniformly on a compact set S, then the sequence is equicontinuous.
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  2. #2
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    Quote Originally Posted by sonictech View Post
    I'm stuck trying to wrap my head around two proofs involving equicontinuity.

    1) If an equicontinuous sequence of functions (fn) converges pointwise to f on a set S, then f is uniformly continuous on S.

    2) If a sequence of continuous functions (fn) converges uniformly on a compact set S, then the sequence is equicontinuous.
    For 1), the definition of equicontinuity says that, given x_0\in S,
    . . . . . . . . . . \forall\epsilon>0\ \exists\delta>0\ \forall n\ |x-x_0|<\delta \Rightarrow |f_n(x)-f_n(x_0)|<\epsilon.
    In that definition, x is a variable point in S, and the absolute value sign denotes the metric in S (which is presumably meant to be a metric space).

    If you let n → ∞ in that definition, and if f_n → f pointwise, then you see that \forall\epsilon>0\ \exists\delta>0\  |x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|\leqslant\epsilon. That shows that f is continuous. It need not be true that f is uniformly continuous. For example, if S=(0,1) and each f_n is the function 1/x then obviously f is also the function 1/x.
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