On the proof that, for every $L^2$-bounded continuous martingale $M$ starting at $0$, the martingale $M^2-[M]$ is uniformly integrable

Question

I am trying to prove that, for every $L^2$-bounded continuous martingale $M$ starting at $0$, the process $M^2-[M]$ is a uniformly integrable martingale. In the proof I am currently reading, they showed that $$E[ \sup_{t\geq0}[M]_t ]=\lim_{t\rightarrow\infty}E[[M]_t]=\lim_{t\rightarrow\infty}E[M_t^2]=E[M_{\infty}^2]<\infty$$ Then, it is clear that $$\sup_{t\geq0}\,(M_t^2-[M]_t)\in L^1\tag{1}$$ Finally they conclude from $(1)$ that $M^2-[M]$ is a uniformly integrable martingale.

However, the point I don't understand is, how uniform integrability can follow from $(1)$. In my opinion, $(1)$ only tells us that $M^2-[M]$ is $L^1$ bounded.

Answer 1

For every $Y$ in $L^1$, the set of random variables $S_Y=\{X\in L^1\mid |X|\leqslant Y\}$ is uniformly integrable.

Apply this fact to $Y=\sup\limits_{t\geqslant0}\,(M_t^2-[M]_t)$ and to the set $\{M_t^2-[M]_t\mid t\geqslant0\}\subseteq S_Y$.