I understand the formal definition for a sequence tending to infinity is:
$$\forall A \gt 0, \quad \exists \, N \in \mathbb{N} \quad such \, that \, \, a_n \gt A, \quad \forall n \gt N. $$
In regards to the part of the definition; $ a_n \gt A, \quad \forall n \gt N$, does that mean that for a certain $N$, $a_N \gt A$, and all values after $N$, hence $n \gt N$ it occurs that, $a_n \gt A$?
There is just one small query I have in applying the definition, using it to prove a trivial sequence:
Prove:$$ a_n =\sqrt n \to\, infinity.$$
Proof: $$Let \quad A \gt 0 \quad be\,\, given.$$
Then, $$\sqrt n \gt A \implies\, n \gt A^2.$$
Intuitively I understand that $N \in \mathbb{N}$ would be chosen so that it is greater than or equal to $A^2$.
But using the above statement:
If $$N=A^2, \,then \, n \gt N \, is \, true.$$
But if $N \gt A^2,$ then could it not occur that $N \gt n$?