Proposition (Cross variation)

Let $M,N$ be continuous local martingales, let $t>0$, let $(\pi_{t}^{n})_{n\in\mathbb{N}}$ be a sequence of subdivisisions on $[0,t]$ with mesh $|\pi_{t}^{n}|\to 0$. Then $$\sum_{i}(M_{t_{i+1}}-M_{t_{i}})(N_{t_{i+1}}-N_{t_{i}})\to [M,N]_{t}$$in probability.