TITLE: Difference of consecutive pairs of sequence terms tends to $0$
QUESTION [8 upvotes]: This seems an elementary problem, but I don't know of any reference to it in the literature.
Consider the sequence $(a_n)_{n=1}^\infty$ of real numbers. Suppose $|a_{n+1}-a_n|\rightarrow0~(n\rightarrow\infty)$, and suppose furthermore that $|a_n|\nrightarrow\infty$. May we conclude that $a_n$ converges?
The second condition precludes the standard example $a_n=\sum_{j=1}^nj^{-1}$, which obviously tends to $+\infty$.

REPLY [5 votes]: $a_i = \sin(\sqrt i)$ oscillates 'forever', never stabilizing, so it is obviously not convergent, however oscillations slow down as $i$ grows: $\sqrt{i+1}-\sqrt i$ decreases with $i$ growing, so does a sine value change $\operatorname{abs}\left(\sin(\sqrt i) - \sin(\sqrt {i-1})\right)$.