Solution
= Solution
Take $N\ge2$ and $q_2=1$. The chain alternates deterministically between zero and one, so $P(X_n=0)$ alternates between one and zero and does not converge.
= Solution
Take $N\ge2$ and $q_2=1$. The chain alternates deterministically between zero and one, so $P(X_n=0)$ alternates between one and zero and does not converge.