If a is an element of R, such that $0\leq a<\epsilon$ for every $\epsilon>0$,then a = 0

Question

"If a is an element of R, such that $0\leq a<\epsilon$ for every $\epsilon>0$,then $a=0$."

Could someone please explain to me as to why this is always true INTUITIVELY? I am able to prove this using proof by contradiction but I still don't understand why this is true intuitively.

Some of the answers to this question might be helpful to your intuition — Ben Grossmann, Oct 01 '14 at 15:27

score 5 · Answer 1 · answered Sep 16 '14 at 16:41

Intuitively, there is no smallest positive real number. (Whatever small positive number you have, you can, for instance, divide it by $2$). The number $a$ described in the problem is smaller than every positive real number. So it can't be positive (or it would be the smallest positive real). But it is greater than or equal to $0$. That leaves $a=0$ as the only possibility.

If a is an element of R, such that $0\leq a<\epsilon$ for every $\epsilon>0$,then a = 0

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