These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it.
My limit and required assumptions:
$\lim \limits_{n \to \infty} \frac{2n + 10}{n} = 2, \epsilon \gt 0, N \in \mathbb{N}, |a_n - a| \lt \epsilon$
Proof: $|a_n - a| = |\frac{2n + 10 -2n}{n}| = \frac{10}{n} \lt \epsilon \rightarrow \frac{1}{n} \lt \frac{\epsilon}{10} \rightarrow n \gt \frac{10}{\epsilon}$
Let $N = [\frac{10}{\epsilon}] + 1$
$n \geq N = [\frac{10}{\epsilon}] + 1 \rightarrow \frac{10}{n} \leq \frac{10}{[\frac{10}{\epsilon}] + 1} \leq \frac{10}{[\frac{10}{\epsilon}]} = \frac{10\epsilon}{10} = \epsilon$
Hence $|a_n - a| \lt \epsilon$
Thank you for your time reading this, and I hope the layout wasn't cryptic! Thank you for any advice!