Conditional expectation given past equals zero implies covariance of zero

Question

Let $X_1, X_2, \ldots$ be random variables on a common probability space (define $X_0 := 0$).

I'd really like the following proposition to be true (and I think that it is) but I'm not sure how to prove it: for all $n \in \mathbb{Z}_{\geq 0}$,

$$ \text{ }\mathbb{E}(X_{n+1} | X_1, …, X_n) = 0 \implies \mathbb{E}(X_i) = Cov(X_i, X_j) = 0
> \text{ for any } i \neq j $$

I suspect that this proof will rest upon some iterated use of the tower property of conditional expectation or some nice property of conditional expectation when conditioning on multiple random variables that I'm not aware of.

Thanks!

Answer 1

For $1 \leq i \leq n$, $E(X_iX_{n+1}|X_1,X_2,...,X_n)=X_iE(X_{n+1}|X_1,X_2,...,X_n)=0$. Take expectation on both sides and finish the proof.