Setup

$(\Omega, F, P)$; probability space

Let $\{F_n\}_{n=0,1,2,\ldots}$ be a filtration, and $X=\{X_n\}_{n=0,1,2,\ldots}$ be an adapted process.In addition, we define$$X^\ast_n = \max_{k\leq n}|X_k|.$$

Then, we want to show that $X^\ast=\{X^\ast_n\}_{n=0,1,2,\ldots}$ is adapted.

My try

It is sufficient to show that$$\{X^\ast_n\in B\}\in F_n$$for $\forall n\in \mathbb{N}\cup\{0\}, \forall B\in\mathcal{B}(\mathbb{R}).$

Now we know$$\{X_n\in B\}\in F_n$$because $X_n$ is adapted.

This is probably incorrect, but it seems to be a good idea to proceed with the transformation in this atmosphere:$$\{X^\ast_n\in B\} = \{\max_{k\leq n}|X_k|\in B\} = \bigcup_{k=1}^n \{|X_k|\in B\}\in F_n$$

I am stuck because I cannot transform it properly.