For any continuous function f (x ) on the interval -1,1, and for any positive integer n, a property of the polynomial pn(x ), which is the best possible approximation to f (x ) in the sense that the maximum absolute value of en(x ) = f (x ) - pn(x ) is as small as possible; namely, that en(x ) assumes its extreme values at least n + 2 times, with the consecutive extrema having opposite signs.