How can I prove the backwards analog for the dominated convergence theorem for conditional expectation

Question

Namely:

Suppose $Y_n \rightarrow Y_{-\infty}$ a.s. as $n \rightarrow \infty$ and $|Y_n|\leq Z$ a.s. where $EZ <\infty$. If $F_n \downarrow F_{-\infty}$ then

$E(Y_n|F_n) \rightarrow E(Y_{-\infty}|F_{-\infty})$ a.s.

This is the dominated convergence theorem for conditional expectations (it's from Durrett)

Also $Y_\infty = E(X|F_{\infty})$