How to prove for every $ε > 0$, it holds that $0 \le a < ε$, then $a = 0$ I have a difficulty on proving this statement; > For every $a$ in $R$ the following holds: > > If for every $ε > 0$ it holds that $0 \le a < ε$, then $a = 0$. I have tried to prove the statement with the **trichotomy** and the technique of proving by contradiction as follows; > Assume that $0 \le a < ε$, then there are 3 possiblities for $a$ and $0$ based on Trichotomy; $a = 0$, $a < 0$ or $a >0$ > > $a<0$ does not hold based on the assumption. So, either $a = 0$ or $a >0$. > > If we multiply $a >0$ by $ε$ on both side; > > $$a ε > 0 ε$$ $$aε > 0$$ Now I have $aε > 0$, and I have no clue to prove that $a = 0$.

You have for **all** $\epsilon > 0$ that $0 \leq a < \epsilon$. With $a$ being fixed assume that $a > 0$ (that is assume that $a\
eq 0$). What happens when $\epsilon = \frac{a}{2}$?