Let x be a sequence in R. Then there exists a subsequence y of x such that y is monotone.

Proof. If x is monotone, then construct any increasing sequence j of integers and then y_n = x_(j_n) is also a monotone sequence. If x is not monotone, then there exists a least integer k_1 such that x_(k_1) is neither an upper or a lower bound for x_M where M > k_1. Now consider the sequence of x beyond x_(k_1); if that sequence attains its minimum, let