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?

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