Limit Superior of a Bounded Sequence of Real Numbers

Let $(a_{n})_{n\in\mathbb{N}}$ be a bounded sequence of real numbers, and for $k\in\mathbb{N}$ write

$$A_{k}=\{a_{m}\;:\;m\in\mathbb{N},\ m\ge k\}.$$

Let $M>0$ be a real number with $|a_{n}|\le M$ for every $n\in\mathbb{N}$, as provided by the boundedness of the sequence, so that $-M\le a_{n}\le M$ for every $n$. Each $A_{k}$ is nonempty, since $a_{k}\in A_{k}$, and is bounded above by $M$, so it has a least upper bound by Least Upper Bound Property of the Real Numbers, unique by Uniqueness of the Supremum and of the Infimum; and the set $\{\sup A_{k}:k\in\mathbb{N}\}$ is nonempty and bounded below, so its infimum exists by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below.

The limit superior of $(a_{n})_{n\in\mathbb{N}}$ is

$$\limsup_{n}a_{n}=\inf\bigl\{\sup A_{k}\;:\;k\in\mathbb{N}\bigr\}.$$