In my Real Analysis class we've been spending some time talking about the $\epsilon$-$N$ definition of convergence. The book we are using, Elementary Analysis by Ross, defines convergence as:
A sequence of real numbers $(s_n)$ is said to converge to $s$ if
For all $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $n\geq N$, $|s_n-s|<\epsilon$.
So he defines $N$ to be a natural number, yet in all the examples of convergence, for example proving that $\lim_{n\to\infty}1/n^2=0$, he says let $N=1/\sqrt{\epsilon}$.
My professor told us that it does not technically matter, so why do we limit ourselves in the definition to the naturals? Wouldn't it just be easier to say there exists an $N$ in the reals? And if we define $N$ to be in the naturals, yet say let $N=1/\sqrt{\epsilon}$, which is clearly not a natural number, why even mention $N$ being a natural number at all?