A bounded sequence cannot be divergent. True or false "A bounded sequence cannot be divergent." Is this statement true?
As far as I know a bounded sequence can either be convergent or finitely oscillating, it cannot be divergent since it cannot diverge to infinity being a bounded sequence. Contrapositively, can I say an unbounded sequence can either be divergent or infinitely oscillating? 

Yes you can.
State that your sequence is unbounded. Suppose it has no upper bound (the same logic is to be applied if it has no lower bound).
So:
$$\forall x \in \mathbb R, \exists n \in \mathbb N, u_n > x$$
Work a bit with this, and the definition of limit:
$$ \exists a \in \mathbb R,  \forall \epsilon \in \mathbb R, \exists n_0 \in \mathbb N, n > n_0 \implies |u_n - a | < \epsilon $$
The definition of limits tells you that for every $\epsilon$, there is a rank $n_0$ such that $(u_n)_n$ is bounded for $n>n_0$. Since there is a finite number of terms $u_n$ for $0 \le n\le n_0$, the sequece is also bounded for $n \le n_0$. 
This is a contradiction with the undounded hypothesis, and you can then conclude that the "convergent" hypothesys is wrong.
