The tail suprema of any real sequence are nonincreasing in $\overline{\mathbb{R}}$, so the limit superior exists for every sequence

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), with tail ranges $T_n$ and extended tail bounds $s_n = \sup T_n$, $i_n = \inf T_n$ as in 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 $\overline{\mathbb{R}}$.

1. Monotonicity of the extended bounds under inclusion. If $A \subseteq B \subseteq \overline{\mathbb{R}}$ (The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined) then $\sup A \le \sup B \qquad \text{and} \qquad \inf B \le \inf A,$ the four quantities being the extended bounds of Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$. No hypothesis is placed on $A$ or $B$; in particular $A$ may be empty.
2. The tail bounds are monotone. $T_m \subseteq T_n$ whenever $n \le m$, and hence $s_m \le s_n \qquad \text{and} \qquad i_n \le i_m \qquad (n \le m).$ In particular $s_{n+1} \le s_n$ and $i_n \le i_{n+1}$ for every $n$, and $i_n \le s_n$ for every $n$.
3. Existence. $\limsup_k x_k$ and $\liminf_k x_k$ exist in $\overline{\mathbb{R}}$ for every sequence of reals, bounded or not.

Claim 1 is the tool the rest of this page uses whenever two extended suprema are compared. It is proved here, from the definition of a least upper bound, rather than quoted from the suprema page, for the reason given in the remarks below.

• The monotonicity is where the two operations of the definition interlock. Because $(s_n)$ is nonincreasing, the outer infimum in $\limsup_k x_k = \inf_n s_n$ is an infimum of a decreasing family, so it is the value the tail suprema are pressing down towards; and because $(i_n)$ is nondecreasing, $\liminf_k x_k = \sup_n i_n$ is the value the tail infima are pressing up towards. Nothing in this lemma says the pressing converges, and for an unbounded sequence there is nothing in $\mathbb{R}$ for it to converge to; the exact statement is For finite $L$: $L = \limsup x_k$ iff for every $\varepsilon > 0$ one has $x_k < L + \varepsilon$ eventually and $x_k > L - \varepsilon$ frequently.

• Why the word "nonincreasing" is spelled out rather than cited. Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences defines monotone for sequences of reals, and $(s_n)$ takes values in $\overline{\mathbb{R}}$, so the definition does not apply to it. Claim 2 is therefore stated as the inequality it is. When $(x_k)$ is bounded every $s_n$ is real (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 $\overline{\mathbb{R}}$) and $(s_n)$ is then a nonincreasing sequence of reals in the sense of Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences, but no proof on this page needs that reading.

• Claim 1 is not Monotonicity of the supremum under inclusion. That lemma is the same one-line argument carried out in $\mathbb{R}$, and its statement carries the hypotheses that the smaller set be nonempty and the larger one bounded above, without which neither supremum denotes anything. Those are exactly the hypotheses that the extended bounds of Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ dispense with, so the extended statement is not an instance of the real one and is proved from the definition of a least upper bound instead.

• Claim 1 costs nothing and is used everywhere. It is the one-line poset argument: the larger set's supremum bounds the smaller set, and leastness does the rest. It is stated as part of this lemma rather than as an item of its own because it is used only in company with the tail bounds.