(Feller Vol.1, P.426, Q.19) Let the chain contain $a$ states and let the state $E_j$ be persistent. There exists a number $q < 1$ such that for $n \ge a$ the probability of the recurrence time of $E_j$ exceeding $n$ is smaller than $q^n$.
Let $f_k$ be the probability of the first return to $E_j$ at the $k$th step. Since $E_j$ is persistent, $\sum_k f_k =1$. I also know that when there are $a$ states, there must be at least one recurrence path in $a$ or less than $a$ steps. This implies that $f_1 + \cdots + f_a > 0$. The probability of the recurrence time of $E_j$ exceeding $n$ is $f_{n+1} + f_{n+2} + \cdots$. This is equal to $$1-(f_1 + \cdots + f_n) < 1-(f_1 + \cdots + f_a) < 1.$$ But, I don't know how to find $q<1$ such that $1-(f_1 + \cdots + f_a) < q^n$. I would appreciate if you give some help.