We give a very elegant (and useful) application of the idea of supremum and infimum. This result is attributed to Michael Fekete, and gets applied in many places, including number theory, combinatorics, and analysis. First, a definition.

Fekete's lemma.

**Proof.**

**Subproof.**

Let

**Subproof.**

Therefore _{
}

Therefore _{
}

For an example of a subadditive sequences,
consider _{
=
}. It is easy to check
_{
+
=
+
+
}.
Fekete's lemma says that (_{
}
** _{
=
}**, for which (