Solution
= Solution
A <reversible Markov chain> must satisfy <detailed balance>. For every $i\geq1$, the rate from $i$ to $i+1$ is positive, while the reverse rate from $i+1$ to $i$ is zero. No positive measure can satisfy $\pi_i g_{i,i+1}=\pi_{i+1}g_{i+1,i}$, so the irreducible chain is not reversible.