A sequence $\{a_n\}_{n=1}^\infty$ is called subadditive if it satisfies the inequality \begin{equation}\label{eqn:seq} a_{n+m}\leq a_n+a_m\qquad\text{for all $n$ and $m$}. \end{equation}The major reason for use of subadditive sequences is the following lemma due to Fekete.

Similarly, a function $f(x)$ is subadditive if \begin{equation*} f(x+y)\leq f(x)+f(y)\qquad\text{for all $x$ and $y$}. \end{equation*}The analogue of Fekete lemma holds for subadditive functions as well.

There are extensions of Fekete lemma that do not require () 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 lemma if some kind of both super- and subadditivity is present. A good exposition of this topic may be found in [2].

References

1

György Polya and Gábor Szegö.
Problems and theorems in analysis, volume 1.
1976.
Zbl 0338.00001.

2

Michael J. Steele.
Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics.
SIAM, 1997.
Zbl 0916.90233.