Sufficiently large

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

<math>f(x)<math> is true for sufficiently large <math>x<math>

which is actually shorthand for:

there exists an <math>a\in\mathbb{R}<math> such that <math>f(x)<math> is true for all <math>x\ge a<math>.

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

"Sufficiently large" is sometimes the subject of mathematical humor; for example, as in the mathematician's joke "π = 3, for sufficiently large values of 3".

Other uses in mathematics

A Haken manifold is sometimes called sufficiently large.