SCHWEITZER 1955 Problem 3

Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$ . Suppose that $D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$ exists. Prove that for every non negative $\lambda$ $P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$ . (P. 7)