Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$? Probability with Martingales:





About $(c)$
My understanding is that $(c)$ is equivalent to:
$$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$
Why is that so?

$$E[\lim A_n] = \lim E[A_n]\ \text{by MCT}$$
$$= \sup E[A_n] \ \because \ P(A_n \le A_{n+1}) = 1 \ \forall \ n \ \because \ P(A_n \le A_{n+1} \ \forall \ n) = 1$$
