Show that that the weak* limit of $\exp(inx)$ is $0$

Question

Let $f_n(x)=\exp(inx)$

I want to show that:

$\hbox{w-}\lim f_n=0$

If I got a correct definition of w-lim I should calculate is follows:

\begin{align*} \hbox{w-}\lim\ f_n &= \lim _{n\to+\infty} \int_{-\infty}^\infty \ f _n (x) \ a(x)\ dx \end{align*}

where $a(\cdot)$ is a fast decreasing function (means that $a(x)$ decreases faster then any Power of $x$?)

I am quite new to distributions.

Answer 1

This is the Riemann-Lebesgue lemma.