For a birth and death process with parameters $\lambda_i$ and $\mu_i$ representing the birth and death rates respectively, define $u_m$ to be the absorption of probability of the process into state $0$, starting from $m$.
The book by Karlin and Taylor mentions that either $u_m=1$ for all $m$ or $u_m\to 0$ as $m\to \infty$. They mention that this fact has a simple probabilistic proof, but I can't seem to get around it.
Any help?