Let $(\Omega,\mathcal{F},P)$ be a probability space, $(\mathcal{F}_n)_n$ a decreasing sequence of sub-$\sigma$-algebra of $\mathcal{F}.$ Let $(X_n)_n$ be a sequence of r.v. converging a.s to $X.$ Prove that if for all $n \in \mathbb{N}, X_n$ is $\mathcal{F}_n$-measurable then $X$ is $\bigcap_{n \in \mathbb{N}}\mathcal{F}_n$-measurable.
We need to show that for a fixed $n \in \mathbb{N},U \in B(\mathbb{R}),X^{-1}(U) \in \mathcal{F}_n. $
Any suggestions is appreciated.