Is infinity always larger than any real number?

Question

Suppose $x$ can be any real number whatsoever. Can it be said that $x < \infty$ is true for every possible $x \in \Bbb R$?

I'm asking the question in the context of infinity as defined by a left open and right unbounded interval in which $x$ is only noted to be a real number. Can it then be said that an $x$ belonging to such an interval will necessarily be smaller then infinity even though the entire real numbers set doesn't actually have a maximal number that you can point to and say that it's smaller then infinity, or perhaps can it just be said that since a right unbounded interval effectively defines its upper bound as infinity, that any $x$ from there will always be smaller then infinity by defintion?

For reference here's the wikipedia page for classification of intervals in which a left open and right unbounded interval is decipted

https://en.wikipedia.org/wiki/Interval_(mathematics)#Classification_of_intervals

Answer 1

This depends on what you mean by $\infty$. If the symbol stands for projective infinity, then no. The projective real line is ordered cyclically. If you mean positive infinity of extended real line ${\overline {\mathbb {R} }}$, then yes.

Answer 2

There are many different types of infinity, but in one way, this is true.

In the “extended real line,” we add to $\mathbb R,$ the real line, the two symbols $+\infty$ and $-\infty,$ and then we extend the ordering so that for every real $x$ we have $-\infty<x<+\infty.$

This is all just definition, or abstract nonsense, but it can be useful.

It does not really help much to understand the concept of the infinite.

It is important to not think of $+\infty$ and $-\infty$ as real numbers. Doing arithmetic with them is generally unwise.