How to prove that infinity ($\forall n \in \mathbb N: x>n$) is not a real number?

Question

By infinity, I mean a number $x$, such that $x>n$ for any natural number $n$. Here, $x + 1$ would just be $x + 1$, so we couldn't use something such as $x + 1 = x > x$ to show that it doesn't exist. Its multiplicative inverse, $\epsilon$, would also be a real number, smaller than any other positive real number.

My first thought was to show that there is no bounded set $S$ such that $x = \inf S$ or $s = \sup S$, but after looking closely at the properties of real numbers, all it says is that if $S$ is a bounded set of reals, then $\sup S$ and $\inf S$ are reals, not necessarily the other way around.

Is my book wrong about this and the converse is also true? Or is there something I'm missing and it can be proved in a different way?

Answer 1

By Archimedean Principle, there exists a natural number $N$ greater than $X$, hence contradiction.

Answer 2

If $\infty \in \mathbb R$ then $\infty$ bounds $\mathbb N$. Show that $\mathbb N$ cannot have a supremum and reach a contradiction.