Let $(a_n)_{n\in\mathbb N}$ be a real sequence and let $D_n:=\{a_k:~k\ge n\}$ be consecutive subsets of its carrier set. Consider the infimum of extended real numbers $\inf D_n$ for each $D_n$. Then $\inf D_n\in \overline{\mathbb R}$ (are elements of the extended real numbers) for all $n$, and the sequence $(\inf D_n)_{n\in\mathbb N}$ is monotonically increasing.
Proofs: 1
Definitions: 1
