Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval).
I know that the following
$ \lim_{\epsilon \to 0_+} \frac{1}{\pi}\int_{-\frac{R}{\epsilon}}^{\frac{R}{\epsilon}}\frac{\sin(z)}{z}(\varphi(\epsilon z)-\varphi(0)) = 0$
is easy to prove assuming the Riemann-Lebesgue lemma, because $\frac{\varphi(x)-\varphi(0)}{x} \in L^1 (\mathbb{R})$ if and only if is in $L^1$ on supp $\varphi$. But can this be also proved in without using the R-L lemma? Any hints appreciated!