Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of random variables, and let $F_n = \sigma(Y_1,Y_2,\dots,Y_n)$ for each $n$.
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables adapted with respect to $\{F_n\}$. i.e. $X_n$ is $F_n$-measurable.
Then, is it true that for each $n$, there exists a function $h_n:\mathbb{R}^{n} \to \mathbb{R}$ such that $X_n = h_n(Y_1,Y_2,\dots,Y_n)$?
Yes. To show it, first see that it is sufficient to prove it in the case when $X_n$ takes the values $0$ or $1$. (Because then you get simple functions, and from there you get everything.) So consider the collection of sets $\theta_n$ which are those sets for which its indicator function is $h_n(Y_1,\dots,y_N)$ for some Borel measurable $h_n:\mathbb R^n \to \{0,1\}$. Now show $\theta_n$ is a sigma field, and that it contains the level sets of $Y_1,\dots,Y_n$. Then conclude that $\theta_n$ contains $\sigma(Y_1,\dots,Y_n)$.