Let \(\left(M_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) and \(\left(N_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) be \(L^{2}\) martingales. Show that \(\left|\langle M, Nangle_{n}ight| \leqslant\) \(\sqrt{\langle Mangle_{n}} \sqrt{\langle Nangle_{n}}\).

Try the classical proof of the Cauchy-Schwarz inequality for scalar products.