A function f(x) is subadditive if In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

for all x and y in the domain of f. In mathematics, the domain of a function is the set of all input values to the function. ...

A sequence { a_n }, n ≥ 1, is called subadditive if it satisfies the inequality This is a page about mathematics. ... In mathematics, an inequality is a statement about the relative size or order of two objects. ...

for all m and n. The major reason for use of subadditive sequences is the following lemma due to Fekete. In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ...

Lemma: For every subadditive sequence { a_n }, n ≥ 1, the limit lim a_n/n exists and is equal to inf a_n/n.

The analogue of Fekete lemma holds for subadditive functions as well.

There are extensions of Fekete's lemma that do not require equation (1) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.¹ A sequence { an }, n ≥ 1, is called superadditive if it satisfies the inequality for all m and n. ...

References

• György Polya and Gábor Szegö. (1976) Problems and theorems in analysis, volume 1, Springer-Verlag, New York. ISBN 0-38705-672-6
• Michael J. Steele (1997) Probability theory and combinatorial optimization, SIAM, Philadelphia. ISBN 0-89871-380-3

Note