Proving convergent sequence theorem.

Take $$a_n = \sum_{k=1}^n\frac 1k.$$ Clearly, $\lim_{n\to\infty}|a_{n}-a_{n-1}| = \lim_{n\to\infty}\frac 1n = 0$, yet $$\lim_{n\to\infty}a_n=+\infty.$$