(3) (15 %) Let $M$ be the normed space of all real-valued continuous functions on $[0, 1]$ with the supremum norm. Let $B \subset M$ such that for any $f \in M$, $f$ is $C^1$ on $[0, 1]$, $f(0) = 0$, and $|f'(x)| \leq 1$. Show that the closure of $B$ in $M$ is compact.