# Fekete’s subadditive lemma

Let $(a_{n})_{n}$ be a subadditive sequence in $[-\infty,\infty)$. Then, the following limit exists in $[-\infty,\infty)$ and equals the infimum of the same sequence:

 $\lim_{n}\frac{a_{n}}{n}=\inf_{n}\frac{a_{n}}{n}$

Although the lemma is usually stated for subadditive sequences, an analogue conclusion is valid for superadditive sequences. In that case, for $(a_{n})_{n}$ a subadditive sequence in $(-\infty,\infty]$, one has:

 $\lim_{n}\frac{a_{n}}{n}=\sup_{n}\frac{a_{n}}{n}$

The proof of the superadditive case is obtained by taking the symmetric sequence $(-a_{n})_{n}$ and applying the subadditive version of the theorem.

Title Fekete’s subadditive lemma FeketesSubadditiveLemma 2014-03-19 22:15:10 2014-03-19 22:15:10 Filipe (28191) Filipe (28191) 4 Filipe (28191) Theorem