Greatest Elements as Suprema, Suprema and Infima of Nested Sets in [−∞,∞], and a Scaling Criterion for Order in [0,∞]

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices:

(i) Let $P$ be a set, $\preceq$ a partial order on $P$ and $A$ a subset of $P$, with suprema formed in $P$ with $\preceq$ as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv). If $g$ is a greatest element of $A$, that is, $g\in A$ and $v\preceq g$ for every $v\in A$, then $g$ is a supremum of $A$, and $\sup A=g$.

Let now $\infty$, $[-\infty,\infty]$, $[0,\infty]$ and the order $\le$ 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), with suprema and infima of subsets of $[-\infty,\infty]$ formed in $[-\infty,\infty]$ with $\le$ as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv); they exist by Order and Suprema in the Extended Real Line (iii).

(ii) If $A\subseteq B\subseteq[-\infty,\infty]$, then $\sup A\le\sup B$ and $\inf B\le\inf A$.

(iii) Let $a,\alpha\in[0,\infty]$, with the product on $[0,\infty]$ as in Addition and Multiplication on [0,∞] (ii). If $c\cdot a\le\alpha$ for every $c\in\mathbb{R}$ with $0<c<1$ (so that $c\in[0,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii)), then $a\le\alpha$.