calculus - Does non-decreasing sequence of this form converge?

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.

Answer

Clearly the the sequence $b_n=a_{n+1}-a_n$ is non-negative, i.e. $b_n\ge0$ for each $n$.

• If any non-negative sequence $b_n\ge0$ is given, can you find a corresponding (non-decreasing) sequence $a_n$ such that $b_n=a_{n+1}-a_n$?


• Can you find a non-negative sequence which does not have limit?