To show the incompleteness of the rational numbers, we just had to find a set of rational numbers, that does not have an supremum / infimum which is element of $\mathbb{Q}$. For example, we could show that the set $\{x\in\mathbb{Q}|x²<2\}$ does not have a supremum in $\mathbb{Q}$, because $\sqrt{2}\notin\mathbb{Q}$. So we simply found a counter example and we were done, right?
Now I'm wondering, how can we show the same thing for $\mathbb{R}$ \ $\mathbb{Q}$ (let's call it $\mathbb{I}$ for simplicity reasons). It feels like, you could just do the same thing we did for $\mathbb{Q}$ also with $\mathbb{I}$, for example we know that $\frac{3}{2}\notin\mathbb{I}$, but how can I actually prove that? In $\mathbb{Q}$ we could atleast construct numbers, but in $\mathbb{I}$ this gets kinda hard. Does someone have a tip for me? Thanks in advance!