Let $B[0,1]$ denote the set of bounded real-valued functions on $[0,1]$. A distance $d$ on $B[0,1]$ is defined by

$$d(f, g)=\sup _{x \in[0,1]}|f(x)-g(x)| .$$

Given that $(B[0,1], d)$ is a metric space, show that it is complete. Show that the subset $C[0,1] \subset B[0,1]$ of continuous functions is a closed set.