HR-MO 2003 Problem 2

The product of the positive real numbers $x, y, z$ is 1. Show that if $$\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z$$then $$\frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k}$$ for all positive integers $k$ .