Why do geometric sets such as $(\infty, x]$ never have infinity included?

Question

I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper way to write this is $(\infty, 3]$, where $3$ is included, as expected, but infinity isn't. Why?

I realize $\infty$ is obviously infinite, and therefore this is hard to imagine, but wouldn't $\infty$ then be included. Otherwise, it seems like the value would NOT include $\infty$.

I can kind of understand, as $\infty-x$ is still infinity. But therefore, should infinity ALWAYS be included? This is a real head-scractcher.

Answer 1

The "half-open" interval you want is $[3, +\infty)$ to denote all values greater than or equal to $3$ : $\{x \in \mathbb{R}, x \geq 3\}$.

You can think of the "open" part of the notation indicating that $+ \infty$ (like $-\infty$) has "no bound".

Answer 2

$-\infty$ and $+\infty$ are not real numbers. By writing for example $(0,+\infty)$ we are referring to all real numbers greater than $0$. In the extended real number system which is the set of all real numbers with $-\infty$ and $+\infty$ adjoined, we have $-\infty<x<+\infty$ for any real number $x$. This explains the use of the interval notation. If we wrote $(0,+\infty]$ instead we would be including $+\infty$ in our set.