If a monotone increasing sequence is not convergent, then is the sequence unbounded?
I know that a monotone increasing sequence that is convergent must be bounded.
But $P \to Q$ being true doesn't necessarily mean $-P \to -Q$ must be true.
Let $(a_n)$ an increasing and divergent sequence of real numbers.
If we suppose that this sequence has an upper bound, being increasing and having an upper bound, it would converge to some real limit : a contradiction. So this sequence doesn't have any upper bound (and in particular, it's unbounded).