Suppose we have a continuous-time Markov chain $X$ on the countably infinite state space $S$. The Markov chain is irreducible and all states are positive recurrent. The transition rates are given by $q(i,j), ~ i \neq j$ and $q(i,i) = - \sum_{j \neq i} q(i,j)$ so that the row sums of the transition rate matrix $Q = (q(i,j))_{i,j \in S}$ are $0$. Define $c(i) := - q(i,i)$.
Let $\pi = (\pi(i))_{i \in S}$ be the limiting probability distribution of $X$.
Is it true that
\begin{equation} \sum_{i \in S} \pi(i) c(i) < \infty \tag{1} \end{equation}
due to the fact that the Markov chain is irreducible and all states are positive recurrent?
In Exercise 2.42 of Liggett, Continuous Time Markov Chains: An Introduction, $(1)$ is assumed to prove that $\pi$ is a stationary distribution if and only if it satisfies $\pi Q = 0$. As I understood it, the solution $\pi$ to $\pi Q = 0$ is a stationary distribution because the continuous-time Markov chain is irreducible and positive recurrent, hence my question.