Chapter 7, Problem 7.24TE

Prove the Cauchy-Schwarz inequality, namely, ${\left(E\left[XY\right]\right)}^2\le E\left[X^2\right]E\left[Y^2\right]$

Hint: Unless $Y=-tX$ for some constant, in which case the inequality holds with equality, it follows that for all $0<E\left[{\left(tX+Y\right)}^2\right]=E\left[X^2\right]t^2+2E\left[XY\right]t+E\left[Y^2\right]$

Hence, the roots of the quadratic equation $E\left[X^2\right]t^2+2E\left[XY\right]t+E\left[Y^2\right]=0$ must be imaginary, which implies that the discriminant of this quadratic equation must be negative.