Let $(a_n)_{n=1}^{\infty}$ be a sequence in $\mathbb{R}$. Suppose that $(a_n)$ is bounded and eventually monotone, meaning that there exists $N \in \mathbb{N}$ such that the tail sequence $(a_n)_{n=N}^{\infty}$ is either nondecreasing or nonincreasing. Then $(a_n)$ converges in $\mathbb{R}$.