A Markov chain with state space $\mathbb{Z}$ is a birth-death chain if the transition probabilities satisfy $p(x,y) = 0$ for $|x-y| > 1$. That is, the only possible transitions are to move one state to the left or right or to stay still.
Suppose such a chain has a stationary distribution $\pi$. It is then a "well-known" fact that $\pi$ satisfies the detailed balance equation $\pi(x) p(x, y) = \pi(y) p(y,x), \quad x,y \in \mathbb{Z}.$ That is, the chain is reversible.
I am looking for a simple proof of this fact (for a class I am teaching). The only proofs I've found go via Kolmogorov's criterion for reversibility, which seems like a lot of work. If possible, the proof should give some insight as to why the particular structure of a birth-death chain causes reversibility to hold.
Thanks!