How can we show $\lim_{n\rightarrow\infty}\int_X\cos(nx) \rightarrow 0\;$?
(X can be any set)
How can we show $\lim_{n\rightarrow\infty}\int_X\cos(nx) \rightarrow 0\;$?
(X can be any set)
Hint Check Riemann–Lebesgue lemma.
Begin by doing it for a closed bounded interval. Then exploit the basic properties of limits.
$\int_X\cos nx\,dx=\left.\frac{1}{n}\sin nx\right|_X\xrightarrow [n\to\infty]{} 0$
as $\,\sin nx\,$ is bounded on $\,X\subset \Bbb R\,$ (why?)