= Solution
Every state reaches state one by deterministic downward steps. From one, the chain can reach a given state $j$ exactly when it can jump to some $k\geq j$ with $\alpha_k>0$ and then count down to $j$. Therefore
$$
\boxed{\text{Irreducible}\quad\Longleftrightarrow\quad\{k:\alpha_k>0\}\text{ is unbounded}.}
$$
If the support is bounded by $M$, states above $M$ cannot be reached from one, so the chain on the stated infinite state space is not irreducible. Its smaller closed class can still have a <stationary distribution>, as in part (e).
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