8 k=1 Show that | Ajk = {X0 = i,X₁ = j i.o.} Hint: show set inclusion both ways. Show that Pi(Ajk) = fij fjjk-¹.

Visiting a state infinitely often. Suppose that X0 = i and consider the event
Ajk defined as the event that the chain visits state j at least k times. Note that if
Ajk occurs, then it must be true that the chain must have made a visit from i to j
and then revisited j at least k − 1 times. Realizing the Pi probability that the chain
visits state j from state i is fij and the Pi probability that the chain visits state j
from state j (k − 1) times is f k−1
jj , we conclude that Pi(Ajk) ≤ fij f k−1
jj . On the other
hand, if the chain happens to visit j from i and also revisit j at least k − 1 times,
then we would know that the event Ajk has occurred.