Suppose we have $(X_n)$ a sequence of real positive random variables in $\mathcal L^2$ that is $\mathcal F_n$ mesurable
Let:
$S_n=X_1+\dots+X_n$ with $S_0=0$
$A_n=\sigma_1^2+\dots+\sigma_n^2$ with $A_0=0$
$V_n=S_n^2-A_n$
Given that $\mathbb E[X_n |\mathcal F_{n-1}]=0$ and $\mathbb E[X_n^2|F_{n-1}]=\sigma_n^2$
Show that $(S_n)$ and $(V_n)$ are martingales.
My problem:
I proved that $(S_n)$ is a martingale simply using the fact that $\mathbb E[X_n |\mathcal F_{n-1}]=0$ but I don't know how am I going to prove that $(V_n)$ is a martingale since the calculation gets complicated if I try to pull out the $X_n^2$ from $S_n^2$ to use the second property.