Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let bounds, suprema and infima of subsets of $\mathbb{R}$ be as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order, bounded and monotone sequences as in Bounded Sequences of Real Numbers §bounded and Monotone Sequences §monotone, and convergence as in Convergent Sequences of Real Numbers §converges. Let $(a_{n})$ be a bounded sequence in $\mathbb{R}$, and for $k\in\mathbb{N}$ let $T_{k}=\{a_{m}:m\in\mathbb{N},\ m\ge k\}$ be the set of its terms from the $k$-th on.

¶ For every $k\in\mathbb{N}$ the set $T_{k}$ is nonempty and bounded, so it has a supremum and an infimum; write $\overline{a}_{k}=\sup T_{k}$ and $\underline{a}_{k}=\inf T_{k}$ for them, the tail supremum and the tail infimum of $(a_{n})$.

¶ $\underline{a}_{k}\le a_{k}\le\overline{a}_{k}$ for every $k\in\mathbb{N}$.

¶ The sequence $(\overline{a}_{k})_{k\in\mathbb{N}}$ of tail suprema is nonincreasing and bounded below, and the sequence $(\underline{a}_{k})_{k\in\mathbb{N}}$ of tail infima is nondecreasing and bounded above.

¶ The infimum $\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and the supremum $\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$ exist, and $\overline{a}_{k}\to\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and $\underline{a}_{k}\to\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$.