\begin{definition}[Definition:Arbitrarily Small]
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''') {{iff}}:
:$\forall \epsilon \in \R_{> 0}: \exists x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$
That is:
:''For any real number $a$, there exists a (real) number not more than $a$ such that the property $P$ holds.''
or, more informally and intuitively:
:''However small a number you can think of, there will be an even smaller one for which $P$ still holds.''
\end{definition}