# Intuition for $\{X>0\}=\cup_{n\in\mathbb{N}}\{X\geq\frac1n\}$?

What is the intuition for the event $$\{X>0\}$$ being equivalent to $$\cup_{n\in\mathbb{N}}\{X\geq\frac1n\}$$? I have seen rigorous expositions on it, but have yet to develop a good inuition of this. Why just the natural numbers, i.e. why not $$\cup_{n\in\mathbb{R}_{\geq0}}\{X\geq\frac1n\}$$, and why don’t we use strict inequalities? When is it useful to write it this way? I have seen it in multiple proofs (e.g. for proving certain quantities are stopping times, proving non-negative r.v. with zero expectation is zero a.s.). but I haven't really seen why only including the natural numbers works. Cheers! (Also, please edit the tags if need be, not sure which ones to include!)

• The equality is still true if you change the natural to the positive reals (not including 0), and it is still true also if you change the relaxed inequality by strict inequality. Have you tried to prove it showing the equality of the sets $\{x\in\mathbb{R}\mid x>0\}$ and $\cup_{n\mathbb N}\{x\in\mathbb R\mid x\geq \frac{1}{n}\}$?
– Javi
Nov 29, 2020 at 14:09

Ultimately, the 'intuition' here is the archimedian property: If $$x >0$$, you can choose $$n$$ so large that $$x > 1/n$$.