Probability of ultimate extinction in a Birth and Death process

Question

The process has infinitely many states, the first one (labelled $0$) absorbing. If $a_n$ is the probability of ultimate extinction given that the process starts in State $n$ (note the corresponding sequence is non-increasing). S. Karlin in his classic book on Stochastic Processes claims that there is a 'simple probabilistic argument' why $\lim_{n\to\infty}a_n$ cannot be a positive number less than $1$ (leaving us with only two possibilities, $0$ and $1$), but does not spell it out (leaving it as an exercise for the reader). What is that argument?