How to prove that $\lim\limits_{n\to\infty} \int_0^1 \cos^n(x)\, dx = 0$ I've got this tasks to prove that:
$\lim\limits_{n\to\infty} \int_0^1 \cos^n(x) \,dx = 0$
I tried to think about a partition ${0,t,1}$, and say that if $t$ is small enough, I can get:
$\lim\limits_{n\to\infty} \int_0^t \cos(x)\, dx = 0$
But then I'm stuck with the rest section $[t,1]$ which approaches $1$.
Any clue?

For any $0<\epsilon< 1$
$$ \int_0^1\cos^n xdx=\int_0^{\frac\epsilon2}\cos^nxdx+\int^1_{\frac\epsilon2}\cos^nxdx\le\frac\epsilon2+\cos^n(\frac\epsilon2)<\frac\epsilon2+\frac\epsilon2=\epsilon\text{ as }n\text{ large}. $$
