Could anyone help me with this?
Let $X_n$ be an irreducible Markov chain on the state space $S=\{1,\ldots,N\}$. Show that there exist $C<\infty$ and $\rho <1$ such that for any states $i,j$,
$ P (X_m \neq j, m = 0, \ldots, n | X_0 =i) \leq C \rho^n.$
Show that this implies that $E(T)<\infty$, where $T$ is the first time that the Markov chain reaches $j$. (Suggestion: There is $\delta >0$ such that for all $i$, the probability of reaching $j$ in some moment of the first $N$ steps, starting at $i$, is major that $\delta$. Why?)
I`ve already got the suggestion but i do not know how to apply it.