I need to prove or disprove these two statements:
If $a_n > 0 $ for all natural numbers $n$, and $a_n$ is not bounded from above, then $a_n → \infty$
If $a_n → +∞ ⇔$ for all $M > 0$, there exist infinitely many terms of $a_n$ larger than M.
For the first one, I was thinking of a contradiction. We assume that the sequence converges to a finite value, and use that to show that there is a contradiction with 'not bounded from above'. Using the fact that $a_n$ > 0, we know that every value must be strictly > 0.
For the second one, I understand the intuition and image in my head, but I can't formulate a sound proof.