Show $g(n)$ takes a finite value for every $n$ for the function $g(n) = \int_{\mathbb{R}} f(x) \sin (nx) \, dx$ where $n \in \mathbb{R}$.
Well, $-1 \le \sin(nx) \le 1$. So if we can prove $f(x)$ is finite then the product of $\sin(nx)$ and $f(x)$ is finite. Would that then make $g(x)$ finite? If so, then how would I go about showing an arbitrary function is finite for each element of its domain?
Could we approximate $g(x)$ by using a function that is dense in $L^1 (\mathbb{R})$ and show that this approximate function is finite for each element in its domain? i.e. approximate f(x) with a simple function, step function, or a continuous function with compact support.
Really stuck here any and all help would be appreciated.