Proof of Real Numbers Lie Strictly below ∞, and Each Element of a Subset of [−∞,∞] Lies below Its Supremum

Throughout, real numbers and $\infty$ lie in $[-\infty,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), and a least element of a class belongs to that class by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iii).

(i): Let $x\in\mathbb{R}$. Then $x\le\infty$ by The Extended Real Line, Its Order, and [0,∞] (iii), since its second argument is $\infty$. Moreover $x\neq\infty$, because $x\in\mathbb{R}$ while $\infty\notin\mathbb{R}$ by The Extended Real Line, Its Order, and [0,∞] (i). Hence $x<\infty$, by the meaning of $<$ in The Extended Real Line, Its Order, and [0,∞] (iii).

(ii): By Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i), $A$ is a subset of the set $[-\infty,\infty]$, so it has a supremum $\sup A$ in $[-\infty,\infty]$ by Order and Suprema in the Extended Real Line (iii). By Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv), $\sup A$ is the least element of the class $\operatorname{Ub}(A)$ of upper bounds of $A$, so $\sup A\in\operatorname{Ub}(A)$. By Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (i) this means that every element of $A$, in particular $a$, satisfies $a\le\sup A$.