How can we show that if $|x| \le 1/n$ for all natural numbers, n, then $x = 0$?

Question

I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]$$

Clearly this should be true, but all my efforts to see so axiomatically have resulted in frustration. So far I have defined the real numbers as a totally ordered field with the axiom of completeness, and have defined the order $a<b$ to mean $0<b-a$

Any explanation, or even a point in the right direction, would be greatly appreciated!

Answer 1

Suposse that $x\neq0$ then $\vert x\vert>0$ and exist $\frac{1}{\vert x\vert}$. For archimedean property there exist a natural $n$ such that $n>\frac{1}{\vert x\vert}\Rightarrow\vert x\vert > \frac{1}{n}$ a contradiction, so $x=0$.