a Martingale $\displaystyle X=(X_n)_{n \geq 0}$ is bounded in $\displaystyle L^2$ if $\displaystyle sup_n E(X_n ^2) \leq \infty$.

Let $\displaystyle X$ be a martingale with $\displaystyle X_n \in L^2$ for each $\displaystyle n$. Show that $\displaystyle X $is bounded in $\displaystyle L^2$ $\displaystyle iff$ $\displaystyle \Sigma_{n \geq 0} ^{\infty} E((X_n-X_{n-1})^2) < \infty$.

here is what i want to know. If we know that $\displaystyle X$ be a martingale with $\displaystyle X_n \in L^2$ for each $\displaystyle n$, it means that $\displaystyle E(X_n ^2) < \infty$ for each $\displaystyle n$, doesnt it? so doesn't this already imply that $\displaystyle X $is bounded in $\displaystyle L^2$?