Definition:Arbitrarily Large

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Let $P$ be a property of real numbers.

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

$\forall a \in \R: \exists x \in \R: x \ge a: P \left({x}\right)$

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