Definition:Arbitrarily Small

Definition

Let $P$ be a property of real numbers.

We say that:

$\map P x$ holds for arbitrarily small $\epsilon$

or:

there exist arbitrarily small $x$ such that $\map P x$ holds

if and only if:

$\forall \epsilon \in \R_{> 0}: \exists x \in \R: \size x \le \epsilon: \map P x$

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.

Also see

• Definition:Sufficiently Small

• Definition:Arbitrarily Large
• Definition:Sufficiently Large

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): arbitrarily large/small