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