The solution is in fact to consider the partial sums of the harmonic series as the terms of the sequence.
Let . Since the harmonic series diverges, a_n diverges.
Clearly, and hence converges to 0.
Prove or give a counterexample: Let an be a sequence such that the sequence bn=an+1 - an converges to 0. Does an have to converge?
Now I know this is not true and I know that it has something to do with partial sums of the harmonic series, but it's been years since I've worked with this sort of material and I am completely lost on how to use the harmonic series to disprove the theorem.