Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $(Y_{n})_{n\in\mathbb{N}}$ be a irreducible Marckov Chain with $Y_{n}:\Omega\rightarrow M$ where $M=\left\{0,1,2,3,4\right\}$ and transition matrix $\Pi$ of size $5\times 5$ and invariant measure $\pi=(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ where $\alpha_{i}\neq 0$ for all $i=0,1,2,3,4$.
Show that the sequence of products $Y_{n}Y_{n-1}\cdots Y_{1}Y_{0}$ eventually vanishes.
Remark: Someone told me this is a consequence of the fact that $\pi(0)=\alpha_{0}\neq 0$.
But I do not see as clearly as $π(0)≠0$ implies that the sequence of products $Y_{0}Y_{1}\cdots Y_{n}$ eventually vanishes.