Hi =)

I just want to check something...

If we have x and a given, and $\displaystyle \forall n \in \mathbb{N}$, $\displaystyle f_n(x) \leq a$ is this inequality true :

$\displaystyle \sup_n f_n(x) \leq a$

?

For those who are interested, it's part of a *simple* proof :

$\displaystyle (f_n)$ is a sequence of measurable functions : $\displaystyle (A, \mathcal{A}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$.

Prove that $\displaystyle \sup_n f_n$ is measurable.

$\displaystyle \square \quad \forall a \in \mathbb{R}, ~ \{\sup_n f_n \leq a\} \overset{(*)}{=} \bigcap_n \{f_n \leq a\}$

$\displaystyle \text{Note that } \{f_n \leq a \}=\{x \in \mathbb{R} ~:~ f_n(x) \leq a\}$

And $\displaystyle \{f_n \leq a \} \in \mathcal{A} \quad \square$

Thanks =)