TITLE: Does non-decreasing sequence of this form converge?
QUESTION [6 upvotes]: Given a non-decreasing sequence $(a_n)$:
$$a_1 \leq a_2 \leq a_3 \leq a_4 \ldots$$
and $$\displaystyle\lim_{n\to\infty}(a_n - a_{n-1}) = 0$$
Does it have to converge?
For strictly than sequence $a_1 < a_2 < a_3 < a_4 < \ldots$ with the limit property, it's easy to show that it doesn't converge, for example take $a_n = \sqrt{n}$. In this case, however I couldn't find a counter example sequence, and I have a feeling this sequence might converge but again I'm not so sure. Any hint would be greatly appreciated.

REPLY [3 votes]: Take for example the harmonic sequence: 
$$ H_n = 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$
It has the property that $H_n \to \infty$, but $H_{n}-H_{n-1}=\frac{1}{n} \to 0$.