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Give an example of an equicontinuous sequence of functions in C[0,1] which has no covergent
subsequence.
Solution
Take f_n(x) = n for all x. For any x, y, and n, |f_n(x) -f_n(y)| = 0, so any delta will
work to show equicontinuity. However, there is clearly no convergent subsequence, since
thevalues of any subsequence at any point go off to infinity.

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Give an example of an equicontinuous sequence of functions in C[0,1].pdf

  • 1. Give an example of an equicontinuous sequence of functions in C[0,1] which has no covergent subsequence. Solution Take f_n(x) = n for all x. For any x, y, and n, |f_n(x) -f_n(y)| = 0, so any delta will work to show equicontinuity. However, there is clearly no convergent subsequence, since thevalues of any subsequence at any point go off to infinity.