Show that
$$ |R_{XY}(\tau)| \le \frac{1}{2}\left[R_X(0) + R_Y(0)\right] $$
I tried use Schwarz's inequality. $$ E^2\{zw\} \le E\{z^2\}E\{w^2\} $$
where z is $z=X(t+\tau)$ and w is $w=X(t+\tau)-Y(t)$. Doing the math, the result was $$ R_{XY}(\tau) \le \sqrt{R_X(0)R_Y(0)} $$ That result is given in many books, but it does not solve my problem. If anyone has any ideas, let me know.