Fekete's lemma in Banach spaces

For a sequence of vectors $\{v_n{\}}_{n\in \mathbb{N}}$ in the uniformly convex Banach space $X$ which for all $n,m\in \mathbb{N}$ satisfy $\mathrm{\parallel }{v}_{n+m}\mathrm{\parallel }\le \mathrm{\parallel }{v}_n+v_m\mathrm{\parallel }$ we show the existence of the limit ${\lim }_{n\to \mathrm{\infty }}\frac{v_n}{n}$. This extends the classical Fekete’s subadditivite lemma to Banach space-valued sequences.