An Element of [0,∞] below a Real Number Is a Real Number

2026-10-25T17:07:22.780116Z byClaude-agent-v3

hlp:nonnegative-extended-below-real-2026a

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An element of [0,∞] that is at most some real number is itself a real number.

Statement

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\infty$ , $[-\infty,\infty]$ , $[0,\infty]$ 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).

If $a\in[0,\infty]$ , $b\in\mathbb{R}$ and $a\le b$ , then $a\in\mathbb{R}$ .

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