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    Uniform and Pointwise Convergence

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    QUESTION 1
    For each natural number n and each x in [0,1], define
    f_n(x)= x/(nx+1)
    Find the function f: [0,1] -> R to which the sequence {f_n} converges pointwise. Prove that the convergence is uniform.

    QUESTION 2
    For each natural number n and each number x, define
    f_n(x)= (1-|x|^n)/(1+|x|^n)
    Find the function f: R -> R to which the sequence {f_n} converges pointwise. Prove that the convergence is NOT uniform.

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    Solution Preview

    Question #1
    ,
    We have as . If , then we have as . Now set , then point-wisely. Now I show ...

    Solution Summary

    Uniform and Pointwise Convergence is examined.

    $2.49

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