Fekete's Subadditive Lemma

Theorem
Let $\sequence {a_n}_{n \ge 1}$ be a subadditive sequence.

Then:
 * $\ds \lim_{n \mathop \to \infty} \frac {a_n} n = \inf_{n \mathop \ge 1} \frac {a_n} n$

Proof
Let $k \ge 1$.

Let $n \ge 1$.

By Division Theorem, there exist $q \in \N$ and $r \in \set {0, 1, \ldots, k-1}$ such that:
 * $n = k q + r$

Thus:

By $n \to \infty$, we obtain:
 * $\ds \forall k \ge 1 : \limsup_{n \mathop \to \infty} \frac{a_n} n \le \frac {a_k} k$

In particular:
 * $\ds (1):\quad \limsup_{n \mathop \to \infty} \frac{a_n} n \le \inf_{k \mathop \ge 1} \frac {a_k} k$

On the other hand, by :
 * $\ds \forall n \ge 1 : \inf_{k \mathop \ge 1} \frac {a_k} k \le \frac{a_n} n$

In particular:
 * $\ds (2):\quad \inf_{k \mathop \ge 1} \frac {a_k} k \le \liminf_{n \mathop \to \infty} \frac{a_n} n$

By Convergence of Limsup and Liminf, $(1)$ and $(2)$ together mean:
 * $\ds \lim_{n \mathop \to \infty} \frac{a_n} n = \inf_{k \mathop \ge 1} \frac {a_k} k$