Show that $X$ is bounded in $L^{2}$ iff $\sum_{n \geq 0} \mathbb{E}\left(\Delta_{n}^{2}\right)<\infty$.

Question

Let $X=\left(X_{n}\right)_{n \geq 0}$ be a martingale in $L^{2},$ let : $$ \Delta_{0}=X_{0}, \quad \Delta_{n}=X_{n}-X_{n-1}, n \geq 1 $$

Show that $X$ is bounded in $L^{2}$ iff $\sum_{n \geq 0} \mathbb{E}\left(\Delta_{n}^{2}\right)<\infty$.

I've already shown that :

$$E(\Delta^2_n) = E(X^2_n)-E(X^2_{n-1})$$ and that $\{\Delta_n\}_n$ is an orthogonal system in $L^2$, I'm not sure if that would help though.

Answer 1

Hint

$$X_n^2=X_0^2+\sum_{i=1}^n(X_{i}^2-X_{i-1}^2)$$