Let $X$ be a Markov chain, with countable state space $I$ and transition probability matrix $P$. $X$ is irreducible, but need not be recurrent. Let $S$ be a fixed subset of $I$.

Define a subset $K$ of $I$ to be "nice" if there exists $\epsilon = \epsilon_K$ such that for all $k \in K$, $P_{kS} \geq \epsilon$. (Here, $P_{kS} = \sum_{s \in S} P_{ks}$.)

Given: with probability 1, there exists a nice set which $X$ visits infinitely often. (Note that the set $K$, and therefore the value of $\epsilon_K$, may be random.)

Want to show: with probability 1, $X$ visits $S$ infinitely often.

It seems like it ought to be either trivially true or trivially false, but I'm failing to determine which...