This is exercise 1.7 of Lawler's "Introduction to stochastic processes."
Let $X_n$ be an irreducible Markov chain on the state space $\{1,\dots,N\}$. Show that there exist $C<\infty$ and $\rho < 1$ such that for any states $i,j$, $$ \mathbb{P}(X_m \ne j, m = 0,\dots,n \mid X_0 = i) \le C \rho^n. $$ Show that this implies $E[T]<\infty$, where $T$ is the first time that the Markov chain reaches the state $j$. (Hint: there exists a $\delta>0$ such that for all $i$, the probability of reaching $j$ after some time in the first $N$ steps, starting at $i$, is greater than $\delta$. Why?)
For the second part, I think the argument is \begin{align*} \mathbb{E}[T] & =\sum_{t=1}^{\infty}\left[t\times\mathbb{P}\{X_{m}\ne j,\forall m\in\{0,\dots,t-1\}\mid X_{0}=i\}\times\mathbb{P}\{X_{t}=j\mid X_{m}\ne j,\forall m\in\{0,\dots,t-1\}\}\right]\\ & \le\sum_{t=1}^{\infty}\left[t\times C\rho^{t-1}\times1\right]=(1-\rho)^{-2}<\infty. \end{align*}
However, I don't understand how to show the inequality (and the Hint). Regarding the hint, the probability is $$ \mathbb{P}(X_m = j, \exists m\in \{0,\dots, N\} \mid X_0 = i) = P_{ij}+P^2_{ij} + \cdots + P^N_{ij}, $$ where $P$ is the transition matrix. Thus, I can take $\frac{1}{2} \min_i \mathbb{P}(X_m = j, \exists m\in \{0,\dots, N\} \mid X_0 = i)$ as $\delta$, but I'm not sure this is the right way to find $\delta$. I see that $$ \mathbb{P}(X_m \ne j, m = 0,\dots,n \mid X_0 = i) = 1 - \mathbb{P}(X_m = j, \exists m\in \{0,\dots, n\} \mid X_0 = i), $$ so I'm guessing that $\delta$ should be of the form $\rho ^n$.
This question is asked on this forum multiple times (1, 2, and 3), but any of these were not helpful.