Sufficiently large

In mathematics, the phrase sufficiently large is used in contexts such as:

P is true for sufficiently large x

which is actually shorthand for:

$\exists~ a\in\mathbb{R}$ such that P is true $\forall~ x\ge a$.

This does not necessarily mean that any particular value for a is known, but only that such an a exists. The phrase "sufficiently large" should not be confused with the phrases "arbitrarily large" or "infinitely large".

Other uses in mathematics

• A sufficiently large manifold is a manifold containing a properly embedded 2-sided incompressible surface.
• Goldbach's conjecture holds only for sufficiently large numbers.

See also

• Mathematical jargon

References

Weisstein, Eric W., "Sufficiently Large" from MathWorld.