Definition:Arbitrarily Large

From ProofWiki
Jump to navigation Jump to search


Let $P$ be a property of real numbers.

We say that $\map P x$ holds for arbitrarily large $x$ (or there exist arbitrarily large $x$ such that $\map P x$ holds) if and only if:

$\forall a \in \R: \exists x \in \R: x \ge a: \map P x$

That is:

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

or, more informally and intuitively:

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

Also see