Prove there exists sequences of random variables $(A_n)$ and $(B_n)$ such that the following hold almost surely for all $n$

Question

Let $(X_n)$ and $(Y_n)$ be sequences of random variables. Suppose that for all $n ≥ 0, X_n = g_n(Y_n)$ for some function $g_n$, and $E[X_{n+1}|Y_n] ≥ X_n$. Prove there exists sequences of random variables $(A_n)$ and $(B_n)$ such that the following hold almost surely for all $n$:

• $A_0 = 0$,
• $A_n ≤ A_{n+1}$,
• $A_n = f_n(Y_{n−1})$ for some function $f_n$, and
• $X_n = B_n + A_n$ and $E[B_n|Y_{n−1}] = B_{n−1}$.

How to prove there exist such sequences of random varible? I couldn't manage to start the prove. Any hint will be appreciated.