Convolution with an approximate identity converges in $L^1$

Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n}\phi=1\), let \(\phi_t(x)\coloneqq t^{-n}\phi(\inverseof{t} x)\) for \(t>0\) be the corresponding approximate identity, and let \(f\in L^1(\RR^n)\). Then \[ \lim_{t\to 0^+}\norm{f * \phi_{t} - f}_1 = 0 . \]