Let $(a_n)_{n=1}^{\infty}$ be a sequence of [real numbers](/page/Real%20Numbers). Then there exists a strictly increasing sequence of indices $(n_k)_{k=1}^{\infty}$ in $\mathbb{N}$ such that the subsequence $(a_{n_k})_{k=1}^{\infty}$ is monotone; that is, $(a_{n_k})_{k=1}^{\infty}$ is either nondecreasing or nonincreasing.