Definition:Sufficiently Large

Definition

Let $P$ be a property of real numbers.

Then:

$P \left({x}\right)$ holds for all sufficiently large $x$

if and only if:

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

That is:

There exists a real number $a$ such that for every (real) number not less than $a$, the property $P$ holds.

It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $a$ actually needs to be known, merely that such a value can be demonstrated to exist.

Also see

• Definition:Arbitrarily Large
• Definition:Sufficiently Small
• Definition:Neighborhood of Positive Infinity