Let $X_n$ be an irreducible Markov chain on the state space $\{1,...,N\}$. Show that there exist $C < \infty$ and $\rho<1$ such that for any states $i,j$, $$\mathbb{P}\{X_m\ne j,m=0,...,n|X_0=i\}\le C\rho^n.$$ Show that this implies that $\mathbb{E}(T)<\infty,$where $T$ is the first time that the Markov chain reaches the state $j$.
This is a question from Introduction to stochastic processes by Lawler. I've considered for a long time but there's nothing occurred to me.
Hint: The identity $\mathbb E(X) = \sum_{k=0}^\infty \mathbb{P}\{X > k\}$ is useful here, as it often is with Markov chains.