Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one can show that the corresponding $\lim_{n\to\infty}a_n$ cannot be positive (thus, can only be equal to 0) 'by a simple probabilistic argument' (claims Karlin, without spelling it out). What exactly is that argument?