it might be a simple question.
I need to prove that $1$ is the supremum of the following set: $A=\left\{\frac{m}{n}\mathrel{}\middle|\mathrel{} m
- $\forall x\in A .x\le 1$
- $\forall \varepsilon>0 \ \exists x\in A .x>1-\varepsilon$
So, the the first requirement is easy to prove, from the defeinition of $A$. but the second is not so simple for me. I've tried to show that this $x$ exists, but I can't show how it looks like.. I guess I should express it with $\varepsilon$, but I have no success so far.
If there are other ways to prove it, it'll fine too, and I'll be happy to hear about them.
Thanks!