Give an example of a bounded, non-convergent real sequence $(a_n)$ s.t. $a_n-a_{n-1}\rightarrow 0$ This was the last part of an exam question. I have spent a long time looking for such an example but have failed (sequences when one of the three conditions is dropped are easily found but I couldn't find one when all three are required). Could you also give the way you thought of the example in your answer please? 
Thanks!

Try $a_n=\sin(\log n)$ for every $n\geqslant1$. Or $a_n=\sin(\sqrt{n})$. Or $a_n=\sin(b_n)$ for any sequence $(b_n)$ such that $b_n\to\infty$ and $|b_{n+1}-b_{n}|\to0$.
