Finite-real supremum and infimum domain

Use the real ordered field of The real numbers and Complete ordered field (least-upper-bound property). For a nonempty set $S\subseteq \mathbb{R}$ bounded above, $\sup S$ denotes the unique real upper bound $u$ such that $u\le v$ for every upper bound $v$. Existence is the least-upper-bound property established for these reals by The reals are complete; if $u,u^{\mathrm{\prime }}$ both satisfy it, $u\le u^{\mathrm{\prime }}$ and $u^{\mathrm{\prime }}\le u$ imply equality. The notation is restricted to this domain.

For nonempty $S$ bounded below, use Greatest lower bound (infimum) and Every nonempty set bounded below has an infimum: $\inf S=-\sup (-S)$, where $-S=\{-s:s\in S\}$. Indeed a lower bound $l$ for $S$ gives the upper bound $-l$ for $-S$; negating its least upper bound gives a lower bound for $S$ greater than every other one.

Neither $\sup \mathrm{\varnothing }$ nor $\inf \mathrm{\varnothing }$ is defined in $\mathbb{R}$: every real is an upper and lower bound, with no least upper or greatest lower bound. If another construction stipulates the value zero for an empty path family, that stipulation is a separate total definition, not $\sup \mathrm{\varnothing }=0$. Infinite values require an explicitly specified extended ordered set; no extended-real field operations are introduced here.