If $x≥0$ and $0≤x0$, then $x=0$? I tried to let ϵ=0.5x, but find ϵ might be zero in this way, which is contradict to for all ϵ>0. Is this statement false?
Prove it by contradiction.

Suppose $x \ge 0$ but $ x \ne 0$. So, $x &gt; 0$

Then for $\epsilon = x/ 2 \gt 0$ we have $ x \not \lt \epsilon$.
You can either take the limit of $\epsilon \to 0$ or suppose that $x \not= 0$ and try to find a contradiction.