Limit superior and limit inferior of a real sequence as ${\inf }_n{\sup }_{k\ge n}{x}_k$ and ${\sup }_n{\inf }_{k\ge n}{x}_k$ in $\bar{\mathbb{R}}$

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences). For $n\in \mathbb{N}$ let

$T_n:=\{x_k:k\in \mathbb{N},k\ge n\}\subseteq \mathbb{R}$

be the $n$-th tail range of $(x_k)$, a nonempty subset of $\mathbb{R}$ since $x_n\in T_n$. Regard $T_n$ as a subset of $\bar{\mathbb{R}}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined) and put

$s_n:=\sup T_n\in \bar{\mathbb{R}},i_n:=\inf T_n\in \bar{\mathbb{R}},$

the supremum and infimum taken in $\bar{\mathbb{R}}$, which exist for every $n$ and for every sequence by Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$. The limit superior and limit inferior of $(x_k)$ are then

${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k:=\inf \{s_n:n\in \mathbb{N}\},{\mathrm{lim\hspace{0.17em}inf}}_k{x}_k:=\sup \{i_n:n\in \mathbb{N}\},$

again taken in $\bar{\mathbb{R}}$ and again existing by Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$, since $\{s_n:n\in \mathbb{N}\}$ and $\{i_n:n\in \mathbb{N}\}$ are subsets of $\bar{\mathbb{R}}$ on which no hypothesis is needed. Both are elements of $\bar{\mathbb{R}}$, and either may be $+\mathrm{\infty }$ or $-\mathrm{\infty }$. The notations ${\mathrm{lim\hspace{0.17em}sup}}_{k\to \mathrm{\infty }}{x}_k$, and ${\stackrel{}{\lim }}_k{x}_k$ all denote the first of them elsewhere; this library writes ${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k$.

Every quantity written here exists, and that is why the extended line was introduced. Each of the four operations above is an application of Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ to a subset of $\bar{\mathbb{R}}$ carrying no hypothesis whatever. Written with the real supremum of Complete ordered field (least-upper-bound property) and the real infimum of Every nonempty set bounded below has an infimum instead, the definition would be available only for sequences that are bounded (Lower bound, bounded below, bounded set): $\sup T_n$ needs $T_n$ bounded above, and $\inf \{s_n\}$ needs $\{s_n\}$ nonempty, bounded below, and made of real numbers (Greatest lower bound (infimum)). None of those is automatic, and the discipline recorded in Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals forbids papering over the gap with a convention. The extended supremum is a different operation in a different ordered set, and it is total.

Values, when the sequence is bounded. If $(x_k)$ is bounded, say $\mathrm{\mid }{x}_k\mathrm{\mid }\le M$ for every $k$, then each $T_n$ is a nonempty subset of $\mathbb{R}$ bounded above by $M$ and below by $-M$, so by the agreement clause of Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ each $s_n$ and each $i_n$ is the real supremum or infimum of $T_n$, and lies in $[-M,M]$. The family $\{s_n\}$ is then a nonempty set of reals bounded below by $-M$, so ${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k$ is likewise the real infimum of $\{s_n\}$ and lies in $[-M,M]$; dually for ${\mathrm{lim\hspace{0.17em}inf}}_k{x}_k$. So for a bounded sequence both quantities are ordinary real numbers computed with the ordinary real supremum and infimum, and the extended line is doing no work. It is only for unbounded sequences that the values $\pm \mathrm{\infty }$ occur.

Sequences of reals: bounded, eventually, frequently, tails, subsequences

The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined

Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$

Complete ordered field (least-upper-bound property)

Every nonempty set bounded below has an infimum

Lower bound, bounded below, bounded set

Greatest lower bound (infimum)

Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals

• The order of the two operations is not symmetric and must be kept straight. $\mathrm{lim\hspace{0.17em}sup}$ is an infimum of suprema and $\mathrm{lim\hspace{0.17em}inf}$ a supremum of infima. Taking them in the other order gives ${\sup }_n{\sup }_{k\ge n}{x}_k={\sup }_k{x}_k$ and ${\inf }_n{\inf }_{k\ge n}{x}_k={\inf }_k{x}_k$, which are the extreme values of the whole sequence and carry no information about its behaviour at large indices. The point of the definition is that the inner operation looks at a tail and the outer one lets the tail recede.

• Why tails at all. Each $s_n$ is a bound on the whole tail from index $n$ on, so it forgets the first $n$ terms; letting $n$ grow forgets any fixed finite number of them. That is what makes $\mathrm{lim\hspace{0.17em}sup}$ and $\mathrm{lim\hspace{0.17em}inf}$ tail quantities in the sense of Convergence depends only on the tail, and it is the reason they can characterise convergence, which is itself a tail property.

• Neither quantity is a limit, and neither is claimed to be one. The symbols $\mathrm{lim\hspace{0.17em}sup}$ and $\mathrm{lim\hspace{0.17em}inf}$ are single pieces of notation for the two displayed expressions, exactly as "$x_k\to +\mathrm{\infty }$" is a single abbreviation in Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$. That the family $(s_n)$ does decrease to ${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k$ in a precise sense is a theorem, not part of this definition; the monotonicity half is The tail suprema of any real sequence are nonincreasing in $\bar{\mathbb{R}}$, so the limit superior exists for every sequence.

• The tail ranges are sets, not sequences. $T_n$ is the range of the $n$-th tail, so repetitions and order are forgotten (Sequences of reals: bounded, eventually, frequently, tails, subsequences). That is harmless here, since a supremum depends only on the set of values, and it is what lets the whole definition be phrased with the order-theoretic operations of Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ and nothing else.