I am unsure how to approach the following question. Given $\{X_1,X_2,...\}$ let $\displaystyle S_n=\sum_{i}^n X_i$ and $F_n=\sigma(X_1,...X_n)$. Suppose that for all $n\geq 1$, $\mathbb E|S_n|<\infty$ and $\mathbb E[S_{n+1}|F_n]=S_n$. Show that $\mathbb E[X_iX_j]=0$ if $i$ does not equal $j$.
Since it is not stated that they are independent how do I go about to approach this question?
Without loss of generality, we assume that $i<j$. Then,
$$X_j = S_j-S_{j-1}$$
entails that
$$\mathbb{E}(X_j \mid \mathcal{F}_i) = \mathbb{E}(S_j \mid \mathcal{F}_i)- \mathbb{E}(S_{j-1} \mid \mathcal{F}_i) = S_i - S_i = 0.$$
Using the tower property, we see that
$$\mathbb{E}(X_i X_j) = \mathbb{E}\big[X_i \mathbb{E}(X_j \mid \mathcal{F}_i) \big] = 0.$$