Let $C^{0,1}([a,b])$ be the space of all Lipschitz-continuous functions $x\colon [a,b] \to \mathbb{R}$ with the metric $ d_{0,1}(x,y) := \sup_{a \le t \le b} |x(t) - y(t)| + \sup_{a \le s,t \le b, s\ne t} \left| \frac{x(s) - x(t)}{s-t} - \frac{y(s) - y(t)}{s - t} \right|. $ So that $x \in C^{0,1}([a,b])$ if $x \in C([a,b])$ and there exists a constant $L$ such that $ |x(s) - x(t)| \le L|s-t| \qquad \forall s,t \in [a,b]. $ Show that
a) The sphere $K = \{ x \in C^{0,1}([a,b]) : d_{0,1}(x,0) \le 1 \}$ is a compact subset of $(C([a,b]), \Delta)$.
b) $C^{0,1}([a,b])$ is not separable.
I knew the definitions, but I have no idea how the attack these problems, do you have any hints?