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How can we show $\lim_{n\rightarrow\infty}\int_X\cos(nx) \rightarrow 0\;$?

(X can be any set)

3 Answers 3

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Hint Check Riemann–Lebesgue lemma.

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Begin by doing it for a closed bounded interval. Then exploit the basic properties of limits.

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$\int_X\cos nx\,dx=\left.\frac{1}{n}\sin nx\right|_X\xrightarrow [n\to\infty]{} 0$

as $\,\sin nx\,$ is bounded on $\,X\subset \Bbb R\,$ (why?)