ELMO 1999 Problem 4

Let $a_1, a_2, a_3, \cdots$ be an infinite sequence of real numbers. Prove that there exists a increasing sequence $j_1, j_2, j_3, \cdots$ of positive integers such that the sequence $a_{j_1}, a_{j_2}, a_{j_3}, \cdots$ is either nondecreasing or nonincreasing.