$\limsup(-x_k) = -\liminf(x_k)$, with the reflection of $\overline{\mathbb{R}}$ exchanging $\pm\infty$

Write $-A := \{-a : a \in A\}$ for $A \subseteq \overline{\mathbb{R}}$, with the reflection of The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined, which fixes no point of $\{-\infty, +\infty\}$ but exchanges the two.

1. Reflection exchanges the extended bounds. For every $A \subseteq \overline{\mathbb{R}}$, $\sup(-A) = -\inf A \qquad \text{and} \qquad \inf(-A) = -\sup A,$ with the 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 $\mathbb{R}$ and no hypothesis on $A$.
2. Reflection exchanges $\limsup$ and $\liminf$. For every sequence $(x_k)$ of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), $\limsup_{k}(-x_k) = -\liminf_{k} x_k \qquad \text{and} \qquad \liminf_{k}(-x_k) = -\limsup_{k} x_k,$ with $\limsup$ and $\liminf$ 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 $\overline{\mathbb{R}}$.

Claim 2 is what turns every statement about $\limsup$ on this page into its dual about $\liminf$ without a second proof, exactly as the identity $\inf S = -\sup(-S)$ does in $\mathbb{R}$. The novelty is only that the reflection now has to move the two new points, and it does: $-(+\infty) = -\infty$.

• Claim 1 needs no hypothesis, and that is the whole gain over $\mathbb{R}$. The corresponding real statement, $\inf S = -\sup(-S)$ (Every nonempty set bounded below has an infimum), carries the hypotheses that $S$ be nonempty and bounded below, because otherwise neither side denotes anything. Here both sides always denote, so the identity is unconditional and can be applied to the family $\{i_n\}$ without first checking that it is bounded, which for an unbounded sequence it is not.

• The reflection is an order anti-isomorphism, not merely a bijection. What step 2.1 uses is that $a \mapsto -a$ is a bijection and reverses the order, both recorded in The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined. A bijection alone would not exchange bounds, and an order-reversing map that is not injective would not carry least upper bounds to greatest lower bounds.

• Consequences on this page. The limit inferior is the least subsequential limit in $\overline{\mathbb{R}}$ is The limit superior is itself a subsequential limit in $\overline{\mathbb{R}}$ and is the greatest one read through this lemma, the $\liminf$ half of $\limsup(x_k + y_k) \le \limsup x_k + \limsup y_k$ whenever the right-hand side is defined in $\overline{\mathbb{R}}$, and dually for $\liminf$ is its $\limsup$ half read the same way, and the $-\infty$ case of A real sequence converges to $L \in \mathbb{R}$ iff $\liminf x_k = \limsup x_k = L$, and diverges to $\pm\infty$ iff both equal $\pm\infty$ is its $+\infty$ case.