Let M be a continuous local martingale starting from $0$. Show that M is an $L^2$-bounded ($\displaystyle \sup_t\|M_t\|_2<\infty$) martingale if $\mathbb{E}([M]_{\infty})<\infty$, where $[M]_{\infty}=\displaystyle \lim_{t \to \infty} [M]_t$ and $[M]$ is the unique continuous adapted non-decreasing process such that $M^2-[M]$ is a continuous local martingale. Could do with some hints.
My thoughts: I know that any local martingale bounded by an integrable random variable is a true martingale and that $\displaystyle \sup_n\mathbb{E}[M^2_{T_n}] < \infty$, where $T_n$ are the stopping times reducing $M^2-[M]$. What I am finding hard is to make any conclusions about $M_t$, given that I have information about $M_{T_n}$ only.
Edit: Managed to show that $M_t \to M_{\infty}$ in $L^2$ for some $M_{\infty}$