A sequence { a_n }, n ≥ 1, is called subadditive if it satisfies the inequality

for all m and n. The major reason for use of subadditive sequences is the following lemma due to Fekete.

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

Similarly, a function f(x) is subadditive if

for all x and y in the domain of f.

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. A good exposition of this topic may be found in [2].

References

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

This article incorporates material from Subadditivity (http://planetmath.org/?op=getobj&from=objects&id=4615) on PlanetMath, which is licensed under the GFDL.