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    Proof that uses limiting values of functions

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    Propose a definition for limit superior lim sup_x-->x_0;x belonging to E of f(x) and limit inferior lim inf_x-->x_0; x belonging to E of f(x) and then propose an analogue of the following for your definition and prove that analogue

    Let X be a subet of R, let f: X-->R be a function, let E be a subset of X, let x_0 be an adherent point of E, and let L be a real number. Then the following two statements are logically equivalent:

    1) f converges to L at x_0 in E
    2) For every sequence (a_n) of n=0 to infinity which consists entirely of elements of E, which convergest to x_0, the sequence (f(a_n)) from n=0 to infinity converges to L.

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    Solution Summary

    The expert examines proofs that uses limiting values of functions.