How do i prove if $\{f_n\}$ is a sequence of extended-real measurable functions, then $\sup_n f_n$ is measurable

Question

Rudin-RCA p.15

Let $X$ be a measurable space. Let $\{f_n\}$ be a sequence of extended-real measurable functions on $X$.

How do i prove that $\sup_n f_n$ is measurable?

Rudin uses a criterion to prove this, that is, if for every real $\alpha$, $f((\alpha,\infty])$ is measurable, then $f$ is measurable.

I don't understand why this is sufficient to prove this.. Help me