I would like to solve the following problem, but I cannot find a good approach to it.
Let $X = (X_n)_{n\geq 0}$ be a homogeneous, irreducible and positive recurrent Markov chain with (countable) state space $S$ and stationary distribution $\pi$. For a subset $A\subseteq S$, let $(T_A^{(k)})_{k\geq 1}$ be the sequence of return times to $A$. Show that $\lim_{k\to\infty} \frac{T_A^{(k)}}{k} = \frac{1}{\sum_{j\in A}\pi(j)}.$
I know that this holds for the case $A = \{i\}$ for any $i\in S$ and also read through the proof of it, but still don't see, how to generalise this to general subsets $A\subseteq S$.