Assuming that you are allowing the possibility of convergence to $-\infty$, doesn't this follow from Fekete's Subadditive Lemma? At every point $x$ the sequence $(f_n(x))$ is subadditive so $$\lim f_n(x)/n=\inf f_n(x)/n.$$ I am assuming that this is what you mean when you say "a subadditive sequence of functions".

Assuming that you are allowing the possibility of convergence to $-\infty$, doesn't this follow from Fekete's Subadditive Lemma? At every point $x$ the sequence $(f_n(x))$ is subadditive so $$\lim f_n(x)/n=\inf f_n(x)/n.$$ I am assuming that this is what you mean when you say "a subadditive sequence of functions," but maybe I misunderstood the question.