Let X be a Banach space. Show that $L = \{ f:X \to \mathbb{R}: f \text{ is Lipschitz and } f(0)=0 \}$ with norm:
$$\|f\|_{lip}= \sup \left\{ \frac{|f(x)-f(y)|}{||x-y||}; x \neq y \in X \right\}$$
is Banach space.
I want to get a sequence of Cauchy in $ L $ and show that converges, I'm trying to transform into a sequence of Cauchy in $\mathbb{R}$ but I'm not getting.