Which extended-real operations this library leaves undefined, and where each $\mathrm{lim\hspace{0.17em}sup}$ statement needs the hypothesis

Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals refused, inside $\mathbb{R}$, the conventions $\sup S=+\mathrm{\infty }$ and $\inf \mathrm{\varnothing }=+\mathrm{\infty }$, and promised that a later page needing $\bar{\mathbb{R}}$ would introduce it explicitly as a new object with its own order and its own partial arithmetic. The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined is that introduction, and this page is the one that needed it. Nothing about the real supremum has changed: $\sup S$ and $\inf S$ for $S\subseteq \mathbb{R}$ are still real numbers, still defined only under the nonempty and bounded hypotheses, and the extended bounds 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 $\mathbb{R}$ are a different operation in a different ordered set, agreeing with the real one exactly where the real one is defined.

What is defined. On $\bar{\mathbb{R}}$ this library defines exactly three things: the total order, the reflection $a\mapsto -a$, and the two partial operations $a+b$ and $ab$. The order and the reflection are total. The two operations are not, and the gaps are these.

What is not defined at all. There is no subtraction on $\bar{\mathbb{R}}$, no division, no absolute value and no exponentiation. Where a proof on this page wants $a-b$ it writes $a+(-b)$, which inherits the gap at $\{+\mathrm{\infty },-\mathrm{\infty }\}$; where it wants a quotient it does not write one. In particular the expressions $(+\mathrm{\infty })-(+\mathrm{\infty })$, $(+\mathrm{\infty })\mathrm{/}(+\mathrm{\infty })$ and $0\mathrm{/}0$ do not occur here, not because they are hard but because neither operation exists.

Why the two gaps are gaps. They are the two places where the value is not determined by the sequences involved, so no assignment could be compatible with limits. For the product this is proved: Null times divergent has no rule: $x_k=1\mathrm{/}k$ with $y_k=ck$ gives product limit $c$, and with $y_k=k^2$ gives divergence ↗ exhibits a null sequence and sequences diverging to $+\mathrm{\infty }$ whose products behave differently, so $0\cdot (+\mathrm{\infty })$ has no value that would make a product rule true. For the sum the same is visible with $a_k=k$ and $b_k=-k$, whose sum is constantly $0$, against $a_k=k$ and $b_k=-2k$, whose sum diverges to $-\mathrm{\infty }$; both pairs have $\mathrm{lim\hspace{0.17em}sup}{a}_k=+\mathrm{\infty }$ and $\mathrm{lim\hspace{0.17em}sup}{b}_k=-\mathrm{\infty }$.

Where each statement on this page carries the hypothesis. Reading the page in order, the pattern is that everything purely order-theoretic is unconditional and everything arithmetic is not.

• 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 $\mathbb{R}$, 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 $\bar{\mathbb{R}}$, The tail suprema of any real sequence are nonincreasing in $\bar{\mathbb{R}}$, so the limit superior exists for every sequence, $\mathrm{lim\hspace{0.17em}sup}(-x_k)=-\mathrm{lim\hspace{0.17em}inf}(x_k)$, with the reflection of $\bar{\mathbb{R}}$ exchanging $\pm \mathrm{\infty }$, $\mathrm{lim\hspace{0.17em}inf}{x}_k\le \mathrm{lim\hspace{0.17em}sup}{x}_k$ for every real sequence, If $x_k\le y_k$ eventually then $\mathrm{lim\hspace{0.17em}sup}{x}_k\le \mathrm{lim\hspace{0.17em}sup}{y}_k$ and $\mathrm{lim\hspace{0.17em}inf}{x}_k\le \mathrm{lim\hspace{0.17em}inf}{y}_k$, The limit superior is itself a subsequential limit in $\bar{\mathbb{R}}$ and is the greatest one and The limit inferior is the least subsequential limit in $\bar{\mathbb{R}}$ carry no hypothesis on the sequence. They use only the order and the reflection, both of which are total, so unbounded sequences and the values $\pm \mathrm{\infty }$ need no separate treatment.
• For finite $L$: $L=\mathrm{lim\hspace{0.17em}sup}{x}_k$ iff for every $\epsilon >0$ one has $x_k<L+\epsilon$ eventually and $x_k>L-\epsilon$ frequently requires $L\in \mathbb{R}$. That is not an arithmetic gap but a syntactic one: its conditions mention $L+\epsilon$ and $L-\epsilon$, and subtraction is not available in $\bar{\mathbb{R}}$. The infinite cases are covered instead by A real sequence converges to $L\in \mathbb{R}$ iff $\mathrm{lim\hspace{0.17em}inf}{x}_k=\mathrm{lim\hspace{0.17em}sup}{x}_k=L$, and diverges to $\pm \mathrm{\infty }$ iff both equal $\pm \mathrm{\infty }$, whose statement uses no arithmetic at all.
• $\mathrm{lim\hspace{0.17em}sup}(x_k+y_k)\le \mathrm{lim\hspace{0.17em}sup}{x}_k+\mathrm{lim\hspace{0.17em}sup}{y}_k$ whenever the right-hand side is defined in $\bar{\mathbb{R}}$, and dually for $\mathrm{lim\hspace{0.17em}inf}$ requires that ${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k+{\mathrm{lim\hspace{0.17em}sup}}_k{y}_k$ be defined, which is exactly the exclusion of the pair $\{+\mathrm{\infty },-\mathrm{\infty }\}$ from the table above. Nothing else is assumed; in particular the sequences need not be bounded, and the two infinite cases are proved rather than excluded.
• For bounded nonnegative sequences, $\mathrm{lim\hspace{0.17em}sup}(x_k{y}_k)\le (\mathrm{lim\hspace{0.17em}sup}{x}_k)(\mathrm{lim\hspace{0.17em}sup}{y}_k)$ requires the sequences to be bounded and nonnegative. Boundedness is what makes all three limit superiors real, so that the product on the right is a product in $\mathbb{R}$; without it the right-hand side could be the undefined $0\cdot (+\mathrm{\infty })$. Nonnegativity is a separate requirement, needed because the estimate multiplies two upper bounds.

What this costs a reader coming from a measure-theory text. Such a text typically declares $0\cdot \mathrm{\infty }:=0$, which is genuinely convenient there, because in an integral the factor $0$ is a measure-zero set and the convention makes countable additivity work without cases. That convention is not in force here, and it is not compatible with limits: it is a decision about a particular formula, not a fact about $\bar{\mathbb{R}}$. A statement quoted from such a source therefore needs its degenerate cases restored before it can be used with the results on this page.

One thing that is not a convention. The equations ${\mathrm{lim\hspace{0.17em}sup}}_k{x}_k=+\mathrm{\infty }$ and ${\mathrm{lim\hspace{0.17em}inf}}_k{x}_k=-\mathrm{\infty }$ occurring throughout this page are ordinary equations between elements of $\bar{\mathbb{R}}$, not abbreviations. That is precisely the difference from Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$, where "$x_k\to +\mathrm{\infty }$" is a single abbreviation for a condition and no object named $+\mathrm{\infty }$ is involved. Both readings coexist without conflict, and A real sequence converges to $L\in \mathbb{R}$ iff $\mathrm{lim\hspace{0.17em}inf}{x}_k=\mathrm{lim\hspace{0.17em}sup}{x}_k=L$, and diverges to $\pm \mathrm{\infty }$ iff both equal $\pm \mathrm{\infty }$ is the statement that relates them.