Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞]

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $-\infty$, $\infty$, $[-\infty,\infty]$ and its order $\le$ be as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), and let suprema and infima of subsets of $[-\infty,\infty]$ be as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv), formed in $[-\infty,\infty]$ with $\le$, which is a total, and so a partial, order on $[-\infty,\infty]$ by Order and Suprema in the Extended Real Line (i); every subset of $[-\infty,\infty]$ has a supremum and an infimum in $[-\infty,\infty]$, by Order and Suprema in the Extended Real Line (iii). Let $a=(a_{m})$ be a sequence in $[-\infty,\infty]$.

(i) For $k\in\mathbb{N}$, the tail infimum and the tail supremum of $a$ at $k$ are

$$\inf_{m\ge k}a_{m}=\inf\{a_{m}:m\in\mathbb{N},\ m\ge k\},\qquad\sup_{m\ge k}a_{m}=\sup\{a_{m}:m\in\mathbb{N},\ m\ge k\},$$

with $m\ge k$ in the order of $\mathbb{N}$ of The Order on Omega: the infimum and the supremum of a subset of $[-\infty,\infty]$, which exist by Order and Suprema in the Extended Real Line (iii).

(ii) The limit inferior of $a$ is the supremum of its tail infima,

$$\liminf_{m\to\infty}a_{m}=\sup_{k}\inf_{m\ge k}a_{m}=\sup\Big\{\inf_{m\ge k}a_{m}:k\in\mathbb{N}\Big\},$$

which exists in $[-\infty,\infty]$ by Order and Suprema in the Extended Real Line (iii).

(iii) The limit superior of $a$ is the infimum of its tail suprema,

$$\limsup_{m\to\infty}a_{m}=\inf_{k}\sup_{m\ge k}a_{m}=\inf\Big\{\sup_{m\ge k}a_{m}:k\in\mathbb{N}\Big\},$$

which exists in $[-\infty,\infty]$ by Order and Suprema in the Extended Real Line (iii).