Irreducible finite-state Markov Chain 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.

Use this fact that if process has not hitted $j$ in $m$ steps, it's not hitted $j$ in $k$ steps where $k$ is greatest multiple of $n$ smaller than $m$.
In fact, if there is a $\delta$ such that beginning from each $i$, you have reached $j$ in $n$ (may be smaller) steps with probablity greater than $\delta$, then take $\rho = (1-\delta)^ {1/n} $  and $C = 1/\rho$. Then, the assertion just says that after $kn+r$ where $(0≤r<n)$ steps, the probablity of not reaching $j$, starting from $i$ is less than $\rho ^ {(k-1)n+r}$. and its obvious. In fact more is true; such probablity is less than $\rho ^ {kn}$ because of strong Markov property.
