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