Epsilon characterisation of the infimum

Given: A nonempty $S\subseteq \mathbb{R}$ that is bounded below, a lower bound $\mathrm{\ell }$ of $S$, and the reflection $T:=-S=\{-s:s\in S\}$.

Reflection, at the level of sets: $S$ is nonempty exactly when $T$ is; $\mathrm{\ell }$ is a lower bound of $S$ exactly when $-\mathrm{\ell }$ is an upper bound of $T$; and $S$ is bounded below exactly when $T$ is bounded above (Reflection through zero exchanges upper and lower bounds). Elementwise, negation reverses the order: $-(-a)=a$, because $(-a)+a=0$ and additive inverses are unique (Field, Identities and inverses in a field are unique); and $a<b$ exactly when $-b<-a$, because translation invariance applied with the constant $-a-b$ turns $a<b$ into $-b<-a$ and, applied with the constant $a+b$, turns it back (Order is preserved by adding a constant and by adding inequalities).

Every nonempty $S\subseteq \mathbb{R}$ bounded below has an infimum, and $\inf S=-\sup (-S)=-\sup T$ (Every nonempty set bounded below has an infimum).

Epsilon characterisation of the supremum: for a nonempty $X\subseteq \mathbb{R}$ bounded above and an upper bound $v$ of $X$, one has $v=\sup X$ if and only if for every $\epsilon >0$ there is $x\in X$ with $v-\epsilon <x$ (Epsilon characterisation of the supremum).