The problem statement, all variables and given/known data Let a_1, a_2, ... be a sequence of real numbers that converges to a. Let b_n = 1/n(a_1 + ... + a_n). Prove that b_1, b_2, ... also converges to a. The attempt at a solution Let e > 0. We know that there is an N such that |a_n - a| < e for all n > N. Now |nb_n - na| <= |a_1 - a| + ... + |a_n - a|, but I don't see how we can have |a_i - a| < e for i = 1, ..., n. Any tips?