# Definition:Arbitrarily Small

## Definition

Let $P$ be a property of real numbers.

We say that $P \left({x}\right)$ holds for arbitrarily small $\epsilon$ (or there exist arbitrarily small $x$ such that $P \left({x}\right)$ holds) if and only if:

$\forall \epsilon \in \R_{> 0}: \exists x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$

That is:

For any real number $a$, there exists a (real) number not more than $a$ such that the property $P$ holds.

However small a number you can think of, there will be an even smaller one for which $P$ still holds.