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Math Help - convergence pointwise

  1. #1
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    convergence pointwise

    Give an example of a sequence of bounded functions f_n converging pointwise to an unbounded function f.
    Prove that if all functions f_n are bounded and f_n converge to f uniformly on a set I then f is bounded.

    Can anybody give me some hints how to prove this question please?

    Thanks for your time indeed
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Give an example of a sequence of bounded functions f_n converging pointwise to an unbounded function f.
    On the open unit interval (0,1), the functions f_n(x) = \frac n{nx+1} are bounded, but converge pointwise to 1/x, which is unbounded.

    Quote Originally Posted by knguyen2005 View Post
    Prove that if all functions f_n are bounded and f_n converge to f uniformly on a set I then f is bounded.
    In words, if f_n converges uniformly to f then for some n, f_n(x) will be within distance 1 of f(x), for all x. But if |f_n(x)| is bounded by M, then |f(x)| will be bounded by M+1.
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