Prove that this Markov chain is irreducible if and only if there exist infinitely many $k\geq0$ such that $q_{k}>0$

Question

We consider the Markov chain $\left(X_{n}\right)_{n\geq0}$ on $\mathbb{N}$ whose transition matrix $P=\left(p_{k,l}\right)_{k,l\geq0}$ given by

$$\forall k\in\mathbb{N} p_{k,k-1}=q_{k},p_{k,k+1}=p_{k}$$

with $p_{k}+q_{k}=1$ and $p_{k}>0$ for any $k\geq0$ .

Prove that this Markov chain is irreducible if and only if there exist infinitely many $k\geq0$ such that $q_{k}>0$ .

How to prove it? I need some hints. Thanks a lot.

Answer 1

This is false. The chain is irreducible only if all $q_k\gt0$. If $q_j=0$, there is no way to get from state $j$ to states $\lt j$.