Archimedean property How can we understand where to apply the Archimedean property of real numbers for a proof or a problem ?

You need it to prove $\lim_{n\to\infty}\frac 1n=0$, for example. You need to have some $n_0$ such that $\frac 1 n<\varepsilon$ when $n\geq n_0$. But $\frac 1{n_0}<\varepsilon$ if and only if $n_0\varepsilon > 1$. Why does such positive integer exist?