For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
For $k \geqslant 1$, show that if $\varphi \in \mathcal{C}_{c}^{k+1}(\mathbb{R})$, we have
$$\left\|(T_{\mu}\varphi)^{(k)} - \varphi^{(k)}\right\|_{\infty} \leqslant \frac{\mu}{2} \left\|\varphi^{(k+1)}\right\|_{\infty}.$$