I am currently taking a real analysis course where we are proving limits. A theorem we are using, precise definition of a limit at infinity, states that if $\varepsilon > 0$ then there exists an $N$ s.t. that $n>N$ implies $|s_n - s| < \varepsilon.$ We use this theorem quite often, however the notation "$n > N$" confuses me. As we've moved on to more difficult proofs "$n > N$" has been occuring more and more. For example, proving
$$\liminf \left|\frac{s_{n+1}}{s_n}\right| < \liminf|s_n|^{1/n} < \limsup |s_n|^{1/n} < \limsup \left|\frac{s_{n+1}}{s_n}\right|$$
Eventually the proof shows $n - N$ fractions exists in $|s_n|.$ However, the meaning of this is lost on me. I always thought $n$ was just some number in the set and $N$ represented the whole set. If that is so then how can $n > N$? I've tried finding the interpretation online and in the book but haven't been able to find anything.
"I always thought [...] $N$ represented the whole set": no. $\mathbb{N}$ is standard notation the set of natural integers. $N$ is not; $N$ is commonly used for some integer, the same way $n$ is used (and, quite often, other letters like $m,k,M,K$...).