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

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 $\lt j$.
