Martingale in $L^2$ equality, williams, probability with martingale p111

Question

I am confused how equality of (e) can be deduced from (f) and (d). I was thinking of applying reverse Fatou's lemma, but (f) doesn't really fit in. Any ideas?

Answer 1

Here's one way to think of it. Let $M'_\infty$ be the $\mathcal L^2$ limit of $M_n.$ Using the $\mathcal L^2$ triangle inequality on the $r\to\infty$ limit of (d) gives (e) with $M'_\infty$ instead of $M_\infty$ and equality instead of inequality. But (f) says $M_\infty=M'_\infty$ as an element of $\mathcal L^2.$