We've begun limits, but the more rigorous approach I have not fully understood yet, so I'm hoping someone could help me please

Question

$\displaystyle lim_{n\rightarrow{\infty}} \frac{n^2+1}{n^2+2}$

Answer

Determine some real number $\displaystyle L$ such that

$\displaystyle |\frac{n^2+1}{n^2+2}-L|<\epsilon$

Rough work(aka scrap work)

BY intuitive guess, L=1. So we need

$\displaystyle |\frac{n^2+1}{n^2+2}-1|<\epsilon$

We need a $\displaystyle N$ such that $\displaystyle \frac{n^2+1}{n^2+2}<\epsilon+1$

Can I choose $\displaystyle N>\frac{1}{1+\epsilon}$, (by Archimedean property),

invert to obtain $\displaystyle \frac{1}{N}<\epsilon+1$

and proceed from there?