(7) \( (10\%) \) Let \( B = \{f\in C([0,1],\mathbb{R}):f \) is \( C^1 \) on \( [0,1] \), \( f(0) = 0 \), and \( |f'(x)|\leq 1\} \). Show that the closure of \( B \) in \( C([0,1]) \) with the supremum norm is compact.