In Rudin $7.14$, Rudin says
Theorem $7.9$ can be rephrased as follow: a sequence $\{f_n\}$ converge to $f$ with respect to the metric $\mathscr C(X)$ if and only if $f_n \rightarrow f$ uniformly on $X$
Theorem $7.9$ states as follow:
Put $$M_n = \sup_{x\in E} |f_n (x) - f(x)|.$$ Then, $f_n \rightarrow f$ uniformly if and only if $M_n \rightarrow 0$.
$\mathscr C (X)$ is defined as the set of all the complex-valued, continuous and bounded functions with domain $X$.
What I don't understand is why does theorem $7.9$ can be rephrased as above?
I feel that there is an extra information, namely boundedness of the functions $f_n$ and of the function $f$, in "convergence with respect to the metric $\mathscr C (X)$" that wasn't present in theorem $7.9$.
The only attempt I can give at figuring this out is that maybe uniform convergence implies boundedness of the functions $f_n$ and $f$ (but I am absolutely not sure of it).
Edit: thanks for your answer guys! I've had been stuck with this all the afternoon!