Infinite Doubly Stochastic Markov Chain

How do we prove that an irreducible, doubly stochastic (row and column sums to 1) discrete time markov chain with infinite state space cannot be positive recurrent?

Initial Ideas: I think the place to start is naturally trying to proving that the stationary distribution cannot possible exist for this setup...but I am having trouble getting started with proving this.

Thanks!

Let $P$ be the transition kernel and let $\nu= \mathbf 1$. Then for each $k$ we have $$(\nu P)_k = \sum_j \nu_jP_{jk} = 1,$$ so $\nu$ is an invariant measure for $P$. Since $P$ is irreducible, any stationary distribution for $P$ must be a multiple of $\nu$. But $\sum_k \nu_k=\infty$, so there cannot be such a distribution.