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    Pointwise limit problem

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    Let fk (x) = x^(1/k), k = 1, 2, 3, . . ..
    a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on the interval [0, 1].
    b) Show that the sequence {fk} infinity k=1 does not converge to f uniformly to f on [0, 1].
    c) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on [sigma, 1], where 0 < sigma < 1.

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    Solution Summary

    Finding the pointwise limit of a sequence of functions. Proving that the sequence does not converge uniformly on the interval [0,1] and converges uniformly on the interval [sigma,1], where 0< sigma< 1.