Someone recently tried to claim that $\lim_{n \to \infty} (a_{n + 1} - a_n) = 0$ implies $\lim_{n \to \infty} a_n$ exists. This is of course not true, as $a_n = \log n$ shows. In fact, even if $a_n$ is bounded the statement isn't true, as the sequence $$0,1,\frac{1}{2}, 0, \frac{1}{3}, \frac{2}{3}, 1, \frac{3}{4}, \frac{2}{4}, \frac{1}{4}, 0, \ldots$$ shows.
What hypotheses can we make on a sequence $a_n$ so that $\lim_{n \to \infty} (a_{n + 1} - a_n) = 0$ implies $\lim_{n \to \infty} a_n$ exists. The well known alternating series test tells us that the statement is true if $a_n$ is the sequence of partial sums of an alternating series. Are there less obscure hypotheses one can make?