How can I prove that a sequence has a given limit? For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?
I'm using the Buchanan textbook, but I'm not understanding their proofs at all.

$$
\lim_{n \rightarrow \infty} \frac {n^2+10}{2n^2} = \lim_{n \rightarrow\infty} \frac {1+\frac {10}{n^2}}{2} = \lim_{n \rightarrow \infty} \left (\frac 12+\frac{10}{2n^2} \right) = \frac 12+5\lim_{n \rightarrow \infty}\frac 1{n^2}=\frac 12
$$
