# Fekete’s subadditive lemma

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

$$\underset{n}{lim}\frac{{a}_{n}}{n}=\underset{n}{inf}\frac{{a}_{n}}{n}$$ |

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 $(-\mathrm{\infty},\mathrm{\infty}]$, one has:

$$\underset{n}{lim}\frac{{a}_{n}}{n}=\underset{n}{sup}\frac{{a}_{n}}{n}$$ |

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 |
---|---|

Canonical name | FeketesSubadditiveLemma |

Date of creation | 2014-03-19 22:15:10 |

Last modified on | 2014-03-19 22:15:10 |

Owner | Filipe (28191) |

Last modified by | Filipe (28191) |

Numerical id | 4 |

Author | Filipe (28191) |

Entry type | Theorem |