$\varepsilon$-$N$ definition of limit to prove that $\lim[(2n+1)/(5n-2)] = 2/5$ as $n$ goes to $\infty$

Question

I've seen this proof before, and it was reduced to $9/(25n-10) < \varepsilon$, and eventually $9/5n < \varepsilon$. When I reduced it, I got $12/(25n - 10) < \varepsilon$, eventually providing:

$$ N > 12/5 \varepsilon. $$

I am wondering if this is correct, and where the $9$ in the numerator of the first proof came from in the first place?

Answer 1

I can't know whether your proof is correct without knowing your whole proof… But here is the source of $9$.

$$ \begin{split} \frac{2n+1}{5n-2} - \frac25 &= \frac{(2n+1) \times 5 - (5n-2) \times 2}{(5n-2) \times 5} \\ &= \frac{(10n+5) - (10n-4)}{25n - 10} \\ &= \frac{9}{25n - 10}. \\ \end{split} $$