<s>[INST] Proving limits without limit theorems I have to prove the following limit without using any limit theorems. I can only do so by using the Archimedean Proprety and the definition of a limit.
I have to prove:
$$\lim_{n\to\infty} \frac{n^2 - 2}{3n^2 + 1} = \frac13$$
 [/INST] The Archimedean Property can be used to show that
$\tag 1 \displaystyle \lim_{n \to \infty}\frac{1}{n^2} = 0$
It is easy to show that
$\tag 2\frac{n^2 - 2}{3n^2 + 1} \lt \frac{1}{3} \text{ for all } n \ge 0$
The following statements are all equivalent for any $0 \lt \varepsilon \lt \frac{7}{3}$ when $n \gt 0$:
$\tag 3 \frac{1}{3} - \varepsilon \lt \frac{n^2 - 2}{3n^2 + 1}$
$\tag , n^2 + \frac{1}{3} - 3 n^2 \varepsilon  - \varepsilon \lt n^2 - 2$
$\tag , \frac{7}{3}  - \varepsilon \lt 3 n^2 \varepsilon$
$\tag 4 \frac{1}{n^2}  \lt \frac{3 \varepsilon}{\frac{7}{3}  - \varepsilon}$
By (1) we can find an $N$ so that (4) is true when $n \ge N$. But then both (2) and (3) show that
$\tag 5 \lim_{n\to\infty} \frac{n^2 - 2}{3n^2 + 1} = \frac13$
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