Order and Suprema in the Extended Real Line

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $-\infty$, $\infty$, $[-\infty,\infty]$, $[0,\infty]$, the order $\le$ and the relation $<$ be as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii). For a subset $A$ of $[-\infty,\infty]$, upper and lower bounds, $\sup A$ and $\inf A$ are as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order, formed in $[-\infty,\infty]$ with $\le$.

(i) $\le$ is a total order on $[-\infty,\infty]$ whose restriction to $\mathbb{R}$ is the order of $\mathbb{R}$: for $x,y\in\mathbb{R}$, $x\le y$ if and only if $x\le y$ in $\mathbb{R}$, and $x<y$ if and only if $x<y$ in $\mathbb{R}$.

(ii) $-\infty$ is the least element and $\infty$ the greatest element of $[-\infty,\infty]$; for $x\in[-\infty,\infty]$, $x\le-\infty$ only if $x=-\infty$, and $\infty\le x$ only if $x=\infty$.

(iii) Every subset $A$ of $[-\infty,\infty]$ has a supremum $\sup A$ and an infimum $\inf A$ in $[-\infty,\infty]$, and $\sup\emptyset=-\infty$ and $\inf\emptyset=\infty$. For a sequence $(u_{n})$ in $[-\infty,\infty]$, $\sup_{n}u_{n}$ denotes $\sup\{u_{n}:n\in\mathbb{N}\}$.

Let $A$ be a subset of $[-\infty,\infty]$.

(iv) If $\infty\notin A$ and $A\cap\mathbb{R}$ is nonempty and bounded above in $\mathbb{R}$, then $\sup A=\sup_{\mathbb{R}}(A\cap\mathbb{R})$, the supremum of $A\cap\mathbb{R}$ formed in $\mathbb{R}$; in particular $\sup A=\sup_{\mathbb{R}}A$ if $A\subseteq\mathbb{R}$ is nonempty and bounded above in $\mathbb{R}$.

(v) If $\infty\in A$, or $A\cap\mathbb{R}$ is not bounded above in $\mathbb{R}$, then $\sup A=\infty$.

(vi) If $-\infty\notin A$ and $A\cap\mathbb{R}$ is nonempty and bounded below in $\mathbb{R}$, then $\inf A=\inf_{\mathbb{R}}(A\cap\mathbb{R})$, the infimum of $A\cap\mathbb{R}$ formed in $\mathbb{R}$.

(vii) If $-\infty\in A$, or $A\cap\mathbb{R}$ is not bounded below in $\mathbb{R}$, then $\inf A=-\infty$.

(viii) $[0,\infty]=\{x\in[-\infty,\infty]:0\le x\}$.

(ix) If $A\subseteq[0,\infty]$, then $\inf A\in[0,\infty]$, and if moreover $A\neq\emptyset$, then $\sup A\in[0,\infty]$.

(x) If $t\in[-\infty,\infty]$ and $t<\sup A$, then there is $a\in A$ with $t<a$. If $t\in[-\infty,\infty]$ and $\inf A<t$, then there is $a\in A$ with $a<t$.