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

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\infty$, $[-\infty,\infty]$, 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), and let suprema of subsets of $[-\infty,\infty]$ be formed in $[-\infty,\infty]$ with $\le$ as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv).

(i) For every $x\in\mathbb{R}$, $x<\infty$.

(ii) Let $A$ be a subclass of $[-\infty,\infty]$ and $a\in A$; then $a\le\sup A$.