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..