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

We prove (i). By The Extended Real Line, Its Order, and [0,∞] (ii), either $a\in\mathbb{R}$, and we are done, or $a=\infty$. Suppose, for a contradiction, that $a=\infty$. By the definition of the order in The Extended Real Line, Its Order, and [0,∞] (iii), $\infty\le b$ means that $\infty,b\in\mathbb{R}$ and $\infty\le b$ in $\mathbb{R}$, or $\infty=-\infty$, or $b=\infty$. By The Extended Real Line, Its Order, and [0,∞] (i), $\infty\notin\mathbb{R}$ and $-\infty\neq\infty$, so the first two alternatives fail; and the third would give $\infty=b\in\mathbb{R}$, again contradicting $\infty\notin\mathbb{R}$. So the case $a=\infty$ cannot occur, and therefore $a\in\mathbb{R}$.