This is computed based on the following recursive formula $$w_n=\frac{\lambda_nw_{n+1}+\mu_nw_{n-1}+1}{\lambda_n+\mu_n}$$ where: $n$ is the inital state, State $0$ is absorbing, $\lambda_n$ and $\mu_n$ are the up and down rates respectively and $$\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}$$diverges (to make extinction certain). To get the recursion started, we need $w_0=0$ and $$w_1=\frac{1}{\mu_1}\sum_{n=1}^\infty\prod_{j=2}^n\frac{\lambda_{j-1}}{\mu_j}$$The derivation of the last formula can be found in S. Karlin's classic book "A first course in stochastic processes". The last step of his proof requires showing that $$\lim_{n\to\infty}\prod_{j=1}^n\frac{\lambda_j}{\mu_j}(w_n-w_{n+1)}=0$$To prove that is, according to Karlin, "more involved but still possible". How does one prove that the last limit must equal to $0$?

This is computed based on the following recursive formula $$w_n=\frac{\lambda_nw_{n+1}+\mu_nw_{n-1}+1}{\lambda_n+\mu_n}$$ where: $n$ is the inital state, State $0$ is absorbing, $\lambda_n$ and $\mu_n$ are the up and down rates respectively and $$\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}$$diverges (to make extinction certain). To get the recursion started, we need $w_0=0$ and $$w_1=\frac{1}{\mu_1}\sum_{n=0}^\infty\prod_{j=1}^n\frac{\lambda_j}{\mu_{j+1}}$$The derivation of the last formula can be found in S. Karlin's classic book "A first course in stochastic processes". The last step of his proof requires showing that $$\lim_{n\to\infty}\prod_{j=1}^n\frac{\lambda_j}{\mu_j}(w_n-w_{n+1)}=0$$To prove that is, according to Karlin, "more involved but still possible" (but he does not do it). How does one prove that the last limit must equal to $0$?