Consider the one-dimensional symmetric random walk of Example 4.19, which was shown in that example to be recurrent. Let πi denote the long-run proportion of time that the chain is in state i.

(a) Argue that πi = π0 for all i.

(b) Show that



i πi = 1.

(c) Conclude that this Markov chain is null recurrent, and thus all πi = 0.