Prove that $d$ is a metric on the set $X$.
$d_u(f,g) = \sup\{|f(x) - g(x)|: x \in I \}, X = C(I)$ the set of all continuous functions from the closed bounded interval $I = [a,b]$ to $\mathbb{R}$
I don't understand what difference using the supremum of $|f(x) - g(x)|$ makes. $|f(x) - g(x)|$ will always be a real number so its seems taking the suprmemum of it is pointless. Am I missing something here?