Martingale convergence theorem for $L^2$ Let $(\Omega, F, P)$ be probability space with probability measure $P$.

Theorem
Let $X\in L^1(P)$, let $F_k$ be an increasing family of sigma algebras, $F_k \subset F$ and $F=\cup_{k=1}^{\infty} \sigma(F_k)$. Then,
$$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$
a.e. $P$ and in $L^1(P)$.

I want to use this theorem for $X\in L^2(P).$
First Since $X\in L^2(P)$, $X\in L^1(P)$.
So, $$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$
a.e. $P$ and in $L^1(P)$.
But, I wanna show that $E[X|F_k] \to E[X|F]$ also in $L^2(P)$.
Could you help me?
 A: *

*By fact 6 of this answer, it suffices to establish that the set of random variables $\{Y_n,n\geqslant 1\}$ where 
$$
Y_n=\left(\mathbb E\left[X\mid\mathcal F_n\right]-\mathbb E\left[X\mid\mathcal F\right]\right)^2,
 $$
is uniformly integrable.

*Since $\mathbb E\left[X\mid\mathcal F\right]$ is square integrable, it suffices to establish uniform integrability of $\{Y'_n,n\geqslant 1\}$ where 
$$
Y'_n=\left(\mathbb E\left[X\mid\mathcal F_n\right] \right)^2.
 $$

*By Jensen's inequality, it suffices to establish uniform integrability of $\{Y''_n,n\geqslant 1\}$ where 
$$
Y''_n= \mathbb E\left[X^2\mid\mathcal F_n\right]  .
 $$

*Boundedness in $L^1$ of $\{Y''_n,n\geqslant 1\}$ is clear; if $A$ is a measurable set, then for all $R$, 
$$
\mathbb E\left[\mathbb E\left[X^2\mid\mathcal F_n\right]\mathbf 1_A\right]=
\mathbb E\left[\mathbb E\left[X^2\mathbf 1\{X^2\leqslant R\}\mid\mathcal F_n\right]\mathbf 1_A\right]+\mathbb E\left[\mathbb E\left[X^2\mathbf 1\{X^2> R\}\mid\mathcal F_n\right]\mathbf 1_A\right]\leqslant R\Pr(A)+\mathbb E\left[ X^2\mathbf 1\{X^2> R\} \right].
$$
A: By Jensen's inequality
$$(E[X|F_n])^2 \leq E[X^2|F_n]$$
Taking Expectation
$$E(E[X|F_n])^2 \leq E[X^2]$$
Then $\sup _{n\geq 0}E(E[X|F_n])^2 < \infty$
By, Martingale $L^p$ convergence Theorem,
$E[X|F_n] \to E[X|F]$ almost surely and in $L^2$.
