Number Equal to Zero iff Arbitrarily Small

Theorem
Let $a \in \R$.

Then:
 * $a = 0 \iff \forall \epsilon \in \R_{>0} : 0 \leq a < \epsilon$

Proof
Let $a = 0$. Then for any $\epsilon \in \R_{>0}$, $0 \leq a < \epsilon$.

Let $a \in \R: a \neq 0$.

If $a < 0$ then $0 \leq a$ does not hold.

Otherwise, $a > 0$ and $a/2 \in \R_{>0}$, but $a < a/2$ does not hold.