Here is the other question I wanted to ask. Say this time I have to prove that
$$\lim_{n\to +\infty} 1+n^2 = \infty$$
This time it's true. The definition of this case reads: $\forall M > 0, \exists N_M > 0$ such that when $n > N_M$ we have $f(n) > M$.
Here
$$1+n^2 > M \implies n > \sqrt{M-1}$$
hence it suffices to take $N_M = \sqrt{M-1}$
- Isn't this a violation of the statement? I mean $\forall M > 0, \exists N > 0$. Not true: indeed for $M = 1$, we have $N_M = 0$.