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Math Help - Uniformly Bounded Sequence

  1. #1
    Dec 2006

    Uniformly Bounded Sequence

    I need some help with this one.

    A sequence of functions f_n is uniformly bounded on a set E iff there is an M>0 s.t. abs(f_n(x)) <= M for all x in E and n in N. Suppose each f_n is a bounded function and f_n converges to F uniformly on E.

    Prove {f_n}is uniformly bounded on E and f is a bounded function on E.


    Since f_n is bounded we know that there exists m<= f_n <=M and we know since f_n converges to f uniformly then for all ε>0 there exists an N in N, s.t. n ≥ N implies (f_n(x)-f(x)) < ε for all of x in E. Then f_n converges and is therefore bounded.

    Since f_n <= M then f_n is uniformly bounded.

    I'm not sure where to go from here.

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  2. #2
    Global Moderator

    Nov 2005
    New York City
    |f(x)-f_n(x)|\leq 1 for n\geq N thus 1 - f_n(x) \leq f(x)\leq 1 + f_n(x). Can you finish now?
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