Expectations of martingales

Question

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$;

$$E(M_n M_k) = E(M_k^2)$$

I am not really sure where to start on this question. Maybe we could use the fact that $M_n = E_n(M_{n+1})$ but I am not sure how this would work.

Answer 1

Hint:

$$ {\rm E}[M_n M_k]={\rm E}[{\rm E}[M_nM_k\mid\mathcal{F}_k]]. $$