54. To prove Eq. (7.24), define the following notation:

Image In terms of this notation, write expressions for

(a) the amount of time during the first m transitions that the process is in state i;

(b) the proportion of time during the first m transitions that the process is in state i.

Argue that, with probability 1,

(c) Image

(d) Image.

(e) Combine parts (a), (b), (c), and

(d) to prove Eq.

(7.24).