So I'm working through a proof of the fact that, for $ 1 < p < \infty $, $ S_R f \rightarrow f $ in $ L^p ( \mathbb{R} ) $. Here, $ \displaystyle S_R f (x) = \int_{-R}^R \hat{f} (\xi) e^{2 \pi i x \xi} \ \mathrm{d} \xi $.
Early on we say that if $ f \in \mathcal{S} (\mathbb{R}) $ then $ S_R f \rightarrow f $ in $ L^2 (\mathbb{R}) $, where $ \mathcal{S} (\mathbb{R}) $ is the Schwartz space. Now, I know that, since $ f $ in $ L^2 $, we have $ \displaystyle f(x) = \int_{-\infty}^{\infty} \hat{f} (\xi) e^{2 \pi i x \xi} \ \mathrm{d} \xi $. Hence we can write $ \displaystyle \vert S_R f (x) - f (x) \vert = \left| \int_{\vert \xi \vert > R} \hat{f} (\xi) e^{2 \pi i x \xi} \ \mathrm{d} \xi \right| $. However I'm struggling to show from this that $ \| S_R f - f \|_2 \rightarrow 0 $.