Let \(\left(f_{n}ight)_{n \geqslant 1} \subset \mathcal{C}_{\infty}\left(\mathbb{R}^{d}ight)\) be a sequence of functions such that \(0 \leqslant f_{n} \leqslant f_{n+1}\) and \(f:=\sup _{n} f_{n} \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}ight)\). Show that \(\lim _{n ightarrow \infty}\left\|f-f_{n}ight\|_{\infty}=0\).