How would you show that for every $\displaystyle x \in \mathbb{R}$, the limit
$\displaystyle \lim_{m\to \infty} (cos(x))^{2m}$
exists?
Use the following facts:
(1) $\displaystyle -1\leq \cos x\leq 1$ for all $\displaystyle x$
(2) If $\displaystyle -1<x< 1$, then $\displaystyle -1<x^2<1$
(3) As $\displaystyle n$ goes to infinity, $\displaystyle x^n$ goes to 0 if $\displaystyle -1<x<1$
Now see if you can put the argument together.