Difficult problem in the analysis of sequences

Question

Let $a_n$ be a sequence of real numbers and let: $$\lim_{n \to \infty}(a_{n+1}-a_n)=0$$ Prove that every $a \in (\lim\text{inf}\,a_n, \lim\text{sup}\,a_n)$ is a limit point of $a_n$.

Answer 1

If you look at $b_n:=a_{n+1}-a_n$ as the terms of a series. You can apply the idea in the proof of the Riemann series theorem. What are the partial sums of $\sum_{n=0}^{\infty}b_n$?