Skorokhod space with uniform norm is Banach

Question

Let $D := D([0,t])$ be the Skorokhod space of right-continuous functions with left limits taking values in $\mathbb{R}^d$. Equip $D$ with the supremum norm $||f||_\infty = \sup_{s \in [0,t]}|f(s)|$. How would one go ahead and show that this is a Banach space? I have thought about that for any Cauchy sequence $(x_n)_{n \in \mathbb{N}}$, then $s \in [0,t]$ $x_n(s)$ converges pointwise to a limit $x$, but I am not sure if that is useful at all. Any hint would be greatly appreciated.

Answer 1

If $(f_n)$ is a Cauchy sequence then $f_n$ is uniformly Cauchy, so there exists $f$ such that $f_n\to f$ uniformly. You only need to show that $f\in\mathcal D$. This follows by exactly the same argument as one uses to show a uniform limit of continuous functions is continuous...