Let $\{X_n\}$ be an irreducible Markov chain with transition probability $P=(p_{ij})$ on a countable state space $S=\{0,1,2,\dots\}$.Suppose $s\in S$.Show that $s$ is a recurrent state if the there is a solution of the following inequality:
$$y_i\ge\sum_{j:j\neq s}p_{ij}y_j \ \ \mbox{for} \ \ i\neq s$$
where $y_i\to \infty $ as $i\to\infty$.