How to prove: If $x \geq 0 $ and $x \leq \epsilon$, for all $\epsilon > 0$, then $x = 0$?

Question

I am trying to prove this problem for my homework. I am having some difficulty with this, because we are just supposed to use several ordered field axioms, the four order axioms, and several basic facts about the real numbers. If anyone can give me help or some guidance that would be much appreciated. :)

Answer 1

Choose $\epsilon = x/2$. What did you get ?

Answer 2

Suppose that $x>0$. Now choose an $\epsilon>0$ such that $\epsilon\in(0,x)$