Let . Pick any epsilon > 0. Then there exists a K such that for any k >= K, (a_(K+1) + ... + a_m) <= S - S_k < epsilon/2 since S_k converges to S and since the partial sums are increasing. Also, since a_n converges to zero, there must exist a J > K such that for all j >= J, (epsilon/2)*min{a_1, a_2, ..., a_K}/S_K >= a_j. Therefore since the sequence is decreasing, for all sufficiently large j, . Since the sequence is positive but eventually remains smaller than any positive number, it must converge to zero.2) and is convergent.

Proove that: