Can I pick any N in the epsilon N - definition? If I were asked to pick a $N$ that would satisfy the definition such that for any $n > N$, the distance between $f(n)$ and the limit would be smaller than a specific epsilon....
...can I just pick $N$'s at random that would satisfy this, and then say that for any $n > N$ it would be true as well? 
I think no; just because it works for a given $N$, doesn't mean it'll work for any $n > N$... but then somebody told me that it would work, because, and he said it like this, "once it's close to the limit, it stays close, and only gets closer", so if the distance between $f(N)$ and the limit is smaller than a given epsilon, then that would remain true for any $n>N$. 
Is this true? Is he making some assumptions that aren't always given? Or is it complete bollocks? 

[In the general case]
Suppose you somehow found some $N$ such that for any $n > N$, we have $|f(n)-L| < \epsilon$.
Then any $N' \ge N$ will also "work," that is, for any $n > N'$, we have $|f(n)-L| < \epsilon$.
However, as Jean-Claude pointed out it is not true that $|f(n)-L|<\epsilon$ implies that for all $n' \ge n$ we have $|f(n')-L| < \epsilon$. But, as I am writing this, you have mentioned that $f$ is monotonic, so in that case, this does hold.
