I am having difficulty understanding a proof that my instructor in real analysis has written for the limit of the sequence $\left(\frac{4n+1}{n+3}\right)$.
She claims that the sequence converges to $4$ if and only if $$\forall\varepsilon>0\ \exists N\ \forall n>N : \left\lvert \frac{4n+1}{n+3} - 4\right\rvert < \varepsilon$$ and as a consequence has said $N=11/\varepsilon-3$ satisfies the definition.
I take issue with this because, according to my research in multiple sources, the definition of a sequence limit demands that $N\in\Bbb N$. Even if I rounded her value up of $N$ up, $\lceil N\rceil$ is only positive if $\varepsilon < 11/3$.
Is my teacher’s proof correct, and $N$ does not have to be a natural number? If her proof is incomplete, how can I complete it?
Strictly speaking, $N$ can be any real number - it's the $n$s which have to be natural numbers. "For every natural number $n>\pi^{1000}$ we have $\vert{4n+1\over n+3}-4\vert<{1\over 2}$" is true, even though our $N$-value (namely $\pi^{1000}$) isn't a natural number itself.
That said, we can always replace a non-natural $N$ with the largest natural $N'<N$ (or $0$ if there is no such natural number), so you don't lose anything by requiring $N\in\mathbb{N}$. The main pedagogical argument in favor of doing this is that it's easier to understand the definition if $n$ and $N$ have the same "type." I'm personally not sure how much I buy this, though: allowing $N$ to be a real number makes it clearer that for a sequence $(s_i)_{i\in\mathbb{N}}$ we have $$\lim_{i\rightarrow\infty}s_i=L\quad\iff\quad\lim_{x\rightarrow\infty}S(x)=L,$$ where $$S(x)=s_{\max\{0,\lfloor x\rfloor\}}$$ is the "step-functionization" of the sequence $(s_i)_{i\in\mathbb{N}}$. The takeaway of course is that limits of sequences are essentially a special case of limits of functions. I think understanding this equivalence demystifies things a bit, since it makes it clear that we really only have a single concept of "limit" floating around here.
(That's not quite true - we're still left distinguishing between limits at infinity and other limits - but that remaining distinction is a harder one to untangle in my opinion.)