Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\tau$ be a measurable map on $(\Omega,\mathcal A,\operatorname P)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$.
$(Y_n)_{n\in\mathbb N}\subseteq L^1(\operatorname P)$ is called subadditive if $$Y_{m+n}\le Y_m+Y_n\circ\tau^m\;\;\;\text{for all }m,n\in\mathbb N\tag1$$ and additive if equality holds.
Assume $(Y_n)_{n\in\mathbb N}$ is subadditive and let $$S_n:=\sum_{i=0}^{n-1}Y_1\circ\tau^i\;\;\;\text{for }n\in\mathbb N,$$ which is easily seen to be additive. Let $$\tilde Y_n:=Y_n-S_n\le0\;\;\;\text{for }n\in\mathbb N,$$ which is additive and nonpositive.
Let $$\mathcal I:=\{A\in\mathcal A:\tau^{-1}(A)=A\}.$$
By Fekete's lemma, $$\operatorname E\left[\frac{\tilde Y_n}n\mid\mathcal I\right]\xrightarrow{n\to\infty}\inf_{n\in\mathbb N}\operatorname E\left[\frac{\tilde Y_n}n\mid\mathcal I\right]\tag2.$$ By Birkhoff's ergodic theorem, $$S_n\xrightarrow{n\to\infty}\operatorname E\left[Y_1\mid\mathcal I\right]\;\;\;\text{almost surely}\tag3.$$
Why are we able to conclude that $\lim_{n\to\infty}n^{-1}Y_n$ exists almost surely?