let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then
(a). The sequence $(a_n)$ may be unbounded.
(b). The sequence $(a_n)$ is bounded but may not converge.
(c). The sequence $(a_n)$ has exactly two limit points.
(d). The sequence $(a_n)$ is convergent.
My work
$$|a_n|-|a_{n+1}|\leq |a_n-a_{n+1}|\leq \frac {n^2}{2^n}$$
$$|a_n|\leq |a_{n+1}|+\frac {n^2}{2^n}$$
So, we can conclude that the sequence is either increasing or decreasing. So, sequece may be unbouded. Is it correct?