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 defines $d_n=a_n-a_{n+1}$ which then leads to$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ where $\alpha$ is a constant, and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (otherwise, $a_n=1$ is the correct solution). To establish the value of $\alpha$, there is a 'simple probabilistic argument' (Karlin) that $\lim_{n\to\infty}a_n$ cannot be positive, thus must be equal to zero (implying that $\alpha=1$). My question is: what is that argument?