"Suppose f_n are defined and continuous on an interval I. Assume that f_n converges uniformly to f on I. If a_n in I is a sequence and a_n -> a, prove that f_n(a_n) converges to f(a)." I don't understand the question. Doesn't uniform convergence imply that for all x in I and e>0, | f_n(x) - f(x) | < e? So in particular, |f_n(a_n) - f(a) | < e.