Task:
$a_n$ is a monotonically increasing sequence, which don't converges.
Proof, that $\lim_{n\to\infty}a_n =+\infty$.
Idea:
We know, if a monotonically increasing sequence is bounded, then it has a limit.
That means, that $a_n$ can't be bounded, otherwise it would be converge.
Is my idea correct?
Best Answer
By definition, $\lim_{n\to\infty} a_n=+\infty$ if for all $M>0$, there exists $N$ such that $n\geqslant N$ implies $a_n > M$.
So let $M>0$. You've shown that $\{a_n\}$ is unbounded, so $M$ cannot be an upper bound for $\{a_n\}$. Therefore there exists an $N$ such that $a_N > M$. Then if $n\geqslant N$, $$a_n\geqslant a_N > M,$$ so that $a_n\to+\infty$.