Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I have tried using the properties:
$\langle aX+bY,Z\rangle = a\langle X,Z\rangle + b\langle Y,Z\rangle$
$\langle X,Y\rangle = \langle Y,X\rangle$
$|\langle X,Y\rangle|^2 \leq \langle X\rangle\langle Y\rangle$
But nothing seems to work. Any help would be appreciated. Thanks.
Expand the inequality $\langle X-Y\rangle\geqslant0$.