Solution
= Solution
For $0\leq k<N$, put
$$
\Delta A_{k+1}=A_{k+1}-A_k
=V_k-\mathbb E[V_{k+1}\mid\mathcal F_k].
$$
This 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 $n\geq1$, so $A$ is previsible; and its nonnegative increments make it nondecreasing.