I am currently reading an article in which the author goes to certain lengths which could be avoided if the following result were true:

Lemma (proposed): Let $T$ be an ergodic measure-preserving transformation of a probability space $(X,\mathcal{F},\mu)$, and let $(f_n)$ be a sequence of integrable functions from $X$ to $\mathbb{R}$ which satisfy the subadditivity relation $f_{n+m} \leq f_n \circ T^m + f_m$ a.e. for all integers $n,m \geq 1$. Suppose that $f_n(x) \to -\infty$ in the limit as $n \to \infty$ for $\mu$-a.e. $x \in X$. Then $\lim_{n \to \infty} \frac{1}{n}\int f_n d\mu <0$.

Via the subadditive ergodic theorem, this effectively states that if $f_n(x) \to -\infty$ almost everywhere then it must do so at an asymptotically linear rate. The supposed lemma would also be equivalent to the statement that if $\frac{1}{n} f_n(x) \to 0$ almost everywhere, then for almost every $x$ the sequence $(f_n(x))$ must return infinitely often to some neighbourhood of $0$ which is not a neighbourhood of $-\infty$. If the sequence $(f_n)$ is additive rather than just subadditive then this last formulation of the result follows from a well-known theorem of G. Atkinson, but the more general subadditive case is less clear.

If the lemma were true then several parts of the paper I am reading would be redundant, which makes me wonder whether it is in fact false. Yet it seems rather plausible. Does anyone know whether this result is true or not?