# Show that $\left( \mathcal{C} ([0,1],\mathbb{R}), ||.||_{\infty} \right)$ is a complete set

Given is the set $$\left( \mathcal{C} ([0,1],\mathbb{R}), ||.||_{\infty} \right)$$. $$||.||_{\infty}$$ denotes the supremum norm. I have to show its completeness and already have a proof ready from my professor who wrote lengty two pages to elaborate on it. My question is: I already know that any space of continous maps from X $$\to$$ Y, that is, $$\left( \mathcal{C} (X,Y)\right)$$ is complete if equipped with the metric $$d_{\infty}(f,g):= sup_{x \in X}\{ d\left(f(x),g(x) \right) \}$$. Hence, since my metric d induces the norm naturally, I should be finished. What am I missing?