Prove $\sum E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converges to $X$ in $L^2$

Question

Let $(X_n)_{n \in \mathbb{N}}$ be a martingale. Prove $\sum_n E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converge to $X$ in $L^2$.

It is not hard at first glance, but I cannot figure it out after many hours .

Add: I assume it is not almost surely converge but can't get a contradiction. Also, I have shown $E(X_{n-1}(X_n−X_{n−1}))=0$, it does not help

Answer 1

First of all, recall that $L^2(\mathbb{P})$ is a complete space; that is $(X_n)_{n \in \mathbb{N}}$ converges in $L^2$ iff it is a Cauchy sequence in $L^2$.

"$\Rightarrow$":

1. Fix $k \leq l$. Using the martingale property show that $$\mathbb{E}((X_{k+1}-X_k) \cdot (X_{l+1}-X_l))=0 \qquad \text{if $k<l$}$$ and $$\mathbb{E}((X_{k+1}-X_k) \cdot (X_{l+1}-X_l)) = \mathbb{E}((X_{k+1}-X_k)^2) \qquad \text{if $k=l$}.$$
2. Deduce from $$X_n-X_m = \sum_{k=m}^{n-1} (X_{k+1}-X_k)$$ and step 1 that $$\mathbb{E}((X_n-X_m)^2) = \sum_{k=m}^{n-1} \mathbb{E}((X_k-X_{k-1})^2).$$
3. Conclude that $(X_n)_{n \in \mathbb{N}}$ is an $L^2$-Cauchy sequence.

"$\Leftarrow$":

1. Recall from step 2 of "$\Rightarrow$" that $$\mathbb{E}((X_n-X_m)^2) = \sum_{k=m}^{n-1} \mathbb{E}((X_k-X_{k-1})^2).$$
2. Deduce that the series $(\sum_{k=0}^n \mathbb{E}((X_{k+1}-X_k)^2))_{n \in \mathbb{N}}$ is a Cauchy sequence (in $\mathbb{R}$).
3. Conclude.