$\mathbf{Riemann-Lebesgue \ Lemma \ in \ L^1(\mathbb{R}^n)}$. Suppose that $f \in L^1(\mathbb{R}^n)$. Then $\hat{f}(k) \rightarrow 0$ as $|k| \rightarrow \infty$.
I cannot understand any of the proofs online, especially this one which seems close to what I would like to prove. https://www.math.ucdavis.edu/~soshniko/201c/hw5sol.pdf. I think a point of particular problems for me is which denseness properties to use as I am not that familiar with these.
If possible can it be done without using compact support as I not seen this concept much? I would as simple a proof as possible.
EDIT: I have checked in the textbook uc davis were using and there is no riemann lebesgue lemma for smoothly compacted functions.
You use the density of smooth compactly supported functions in $L^1(\mathbb{R})$.
Namely, if $f\in L^1(\mathbb{R})$. Then there is a sequence of smooth compactly supported functions $f_n$ such that $\|f-f_n\|_{L^1(\mathbb{R})}\to0$ as $n\to\infty$. This is exactly what lines 6-7 of the solution say.