Let $C[a,b]$ denote all continuous functions $:[a,b] \rightarrow \mathbb{R}$, with the metric
$$d(f,g)=\sup |f(x) - g(x)| \text{ for } a\leq x \leq b.$$
Show that $d(f,g)$ is a metric space.
I have started with this: Since continuos function is bounded, so i use the same proof from it: " $B[a,b]$ denote for all bounded functions $:[a,b] \rightarrow \mathbb{R}$, with
$d(f,g)=\sup |f(x) - g(x)|$ for $a\leq x \leq b$ "
to prove those (continuos function is metric space). Is it right? Or there is another way?