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Math Help - Lim sup of a sequence of functions

  1. #1
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    Lim sup of a sequence of functions

    Try to help me to determine whether the following statement is true or not (if true, prove; if false, give a counterexample):

    Let f_j be a sequence of functions on a domain U, and suppose that \limsup_{j\rightarrow \infty}{f_j(x)}\leq C<\infty for all x\in U, then for each \epsilon>0, there exists a J, such that f_j(x)\leq C+\epsilon for all j\geq J and all x\in U.

    Thanks!
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  2. #2
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    Quote Originally Posted by frankmelody View Post
    Try to help me to determine whether the following statement is true or not (if true, prove; if false, give a counterexample):

    Let f_j be a sequence of functions on a domain U, and suppose that \limsup_{j\rightarrow \infty}{f_j(x)}\leq C<\infty for all x\in U, then for each \epsilon>0, there exists a J, such that f_j(x)\leq C+\epsilon for all j\geq J and all x\in U.

    Thanks!
    Suppose the claim were false. Then we would have f_j(x)>C+\epsilon for infinitely many j. And so we would be able to find a subsequence such that f_{j_k}(x)>C, which would contradict \limsup_{j}f_j(x)\le{C}
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