Prove that $\sup\limits_nf_n$ and $\inf\limits_nf_n$ are measurable.

Question

Let $(X,\cal{A}$) a measurable space and $f_n : X \to [-\infty,+\infty]$ a sequence of measurable functions.
Prove that $\sup\limits_nf_n$ and $\inf\limits_nf_n$ are measurable.

Proof: To prove that a function $f$ is measurable, we have to show that the set $[f \leq b]= \{x \in X : f(x) \leq b\} \in A$ for every $b \in \mathbb{R}$. So

For every $b \in \mathbb{R}$ $$[\sup\limits_nf_n \leq b]=\cap_{n=1}^{\infty}[f_n \leq b] \in \cal{A}$$ and $$[\inf\limits_nf_n \leq b]=\cup_{n=1}^{\infty}[f_n \leq b] \in \cal{A}$$ But i think that it should be: $$[\sup\limits_nf_n \leq b]=\cup_{n=1}^{\infty}[f_n \leq b] \in \cal{A}$$ and $$[\inf\limits_nf_n \leq b]=\cap_{n=1}^{\infty}[f_n \leq b] \in \cal{A}$$.