hi, I'm sorry if the question is silly, but I couldn't get my head around it for a while now.

In Markov Chains (MC) proving that a state is either recurrent or transient is through Borel-Cantelli lemma (BCL): the event (state) happens infinitely often (i.o.) if the series diverges and vice versa.

The question is, how to apply BCL to prove that the state is absorbing.

The problem is that if the state is absorbing (i.e. once we got there we're stuck in it) then, at least intuitively, it is an event that happens almost always (a.a.) rather than i.o., which means that BCL isn't applicable here, at least not directly (since BCL applies only to events that happen i.o.).

1ce again, sorry if the question is too silly.