(short) Proof explanation of indexed unity - why does such n exists

Question

this proof and specifically the first line is throwing me off. How can we assume such a value n_x exists? How would one argue for this? (or does this method not work?)

Answer 1

According to Archimedean Principle for every $0<\epsilon <1$ there exist an $n\in \mathbb{N}$ such that $\frac{1}{n}<\epsilon$. So $1-\frac{1}{n}>1-\epsilon $. But $1-\epsilon $ is arbitrary, Therefore for every $0<x<1$, there is $n_x\in \mathbb{N}$ such that $1-\frac{1}{n_x}>x $