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?
$$\ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \  (-1)^n$$
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:

$$\forall x \in \mathbb R, \exists n \in \mathbb N, u_n &gt; x$$

Work a bit with this, and the definition of limit:

$$ \exists a \in \mathbb R,  \forall \epsilon \in \mathbb R, \exists n_0 \in \mathbb N, n &gt; n_0 \implies |u_n - a | &lt; \epsilon $$

The definition of limits tells you that for every $\epsilon$, there is a rank $n_0$ such that $(u_n)_n$ is bounded for $n&gt;n_0$. 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.