Fekete's lemma with an extra constant in the index

Say I have an "almost" subadditive sequence $(a_n)_{n\in\mathbb{N}}$ in the sense that$$a_m+a_n\geq a_{m+n+k}$$for some fixed $k\in\mathbb{N}$. Then do I have that the limit $\lim_{n\to\infty}\frac{a_n}{n}$ exists and equals $\inf\frac{a_n}{n}$? Thanks in advance.