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.

In general, we say that a sequence $x_n \rightarrow x$ if $|x_n - x| \rightarrow 0$. This means that for all $\epsilon > 0$ there exists some $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - x| < \epsilon$.
We can also use results about sequences that will be proved in any good introductory texts, such as if $x_n \rightarrow x$ and $y_n \rightarrow y$, $\lambda \in \mathbb{R}$: 
$i) x_n + y_n \rightarrow x+y$
$ii) x_ny_n \rightarrow xy$
$iii)\lambda x_n \rightarrow \lambda x$
$iv)$ If $x \not= 0$ then  $\frac{y_n}{x_n} \rightarrow \frac{y}{x}$
Using these results we can see that $\displaystyle c_n = \frac{n^2 + 10}{2n^2} = \frac{1+\frac{10}{n^2}}{2}$. So if $a_n = \frac{10}{n^2} \rightarrow a$ then $c_n \rightarrow \frac{1+a}{2}$. Now we can prove that $a_n \rightarrow 0$ using epsilon delta techniques, so it follows that $c_n \rightarrow \frac12$.
