 Solution

For $0\le k<N$, putThis is ${\mathcal{F}}_k$-measurable, and it is nonnegative because $V$ is a supermartingale. Hence $A_n$, being the sum of increments known by time $n-1$, is ${\mathcal{F}}_{n-1}$-measurable for $n\ge 1$, so $A$ is previsible; and its nonnegative increments make it nondecreasing.