Can I say that a fixed constant is less or equal infinity?

Question

Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$?

E.g., is $10 \leq \infty$ a correct mathematical statement?

I know this comparison is true in computer arithmetic, however is it correct from mathematical point of view? Does the "equality" part in $\leq$ matter here?

Answer 1

The real numbers $\mathbb R$ do not contain $\infty$ as an element, so with the relation $\le_\mathbb R$, the statement $c\le\infty$ does not make sense.

The extended real number line $\overline{\mathbb R}$ does contain both $\infty$ and $-\infty$. While it loses the additive group structure of the standard reals, it retains a total ordering induced by $\le_\overline{\mathbb R}$, and under that relation, it is indeed true that

$$c\le_\overline{\mathbb R}\infty$$

for any $c\in \mathbb R$.

Answer 2

Sure. An example where this is used in mathematics is when talking about $L^p$-spaces. The only thing you need to know about these is that they depend on a real parameter $p$ which is allowed to take the value $\infty$, and it's common to make statements about $L^p$-spaces which are uniform in the parameter $p$, e.g. to say "for all $1 \le p \le \infty$, the $L^p$-space satisfies..."