Consider an irreducible aperiodic (so that it's positive recurrent) Markov chain. Let $\lim_{n \to \infty}p_{ii}(n) =: \pi_{i}$, where $p_{ii}(n)$ denotes the $(i,i)$-element of the $n$th power of the transition matrix, i.e., $P^{n}$. By the fact that $\pi_{i}$ is the reciprocal of the expected return time of state $i$ and that for a positive recurrent state, the expected return time is finite, we know $\pi_{i} > 0.$
It is also a fact that $\lim_{n \to \infty}p_{ij}(n) = \lim_{n \to \infty}p_{jj}(n) = \pi_{j}$ so we have $\sum^{\infty}_{j = 0}\lim_{n \to \infty}p_{ij}(n) = \sum^{\infty}_{j = 0}\pi_{j}$.
Now the professor said by Fatou's Lemma, we have $\lim_{n \to \infty} \sum^{\infty}_{j = 0}p_{ij}(n) \geq \sum^{\infty}_{j = 0}\pi_{j}$, which I can't figure out why. I don't know how I should define the sequence of functions $\{f_{n}\}$ to apply the theorem, and I would appreciate any help!