Let $(F_n)_n$ be a filtration and $F=\sigma(\cup_n F_n)$.
Let $Z$ be a random variable such that $\mathbb{E}Z^2<\infty$.
Let $X_n=\mathbb{E}[Z|F_n]$.
Given:
If $X_n\to Z$ a.s. then $||X_n-Z||_2\to0$ iff $||X_n||_2\to ||Z||_2$.
For a $\pi$-system $\Pi$ such that $\sigma(\Pi)\subset F$, if we have $\int_A Yd\mathbb{P}=\int_A Xd\mathbb{P}$ for all $A\in \Pi$ then
$$Y=\mathbb{E}[X|\sigma(\Pi)].$$
How do we show that $||X_n-Z||_2\to0$?
I know that
$$||X_n||_2=\left(\mathbb{E}X_n^2\right)^{1/2}=\left(\mathbb{E}[\mathbb{E}[Z|F_n]^2]\right)^{1/2}.$$
By conditional Jensen I thought that then $||X_n||_2\leq\left(\mathbb{E}[\mathbb{E}[Z^2|F_n]]\right)^{1/2}=\left(\mathbb{E}[Z^2]\right)^{1/2}=||Z||_2$.
Is this correct? How do I continue from here?