Take any decreasing sequence $(a_n)_{n=0}^\infty$ and let $$\lambda_n:=a_{n-1}-a_n, \quad \mu_n:=a_n-a_{n+1} $$ for natural $n$. Then the sequence $(a_n)_{n=0}^\infty$ is a solution of the equation \begin{equation*} a_n=\frac{\lambda_n a_{n+1}+\mu_n a_{n-1}}{\lambda_n+\mu_n} \tag{1} \end{equation*} for natural $n$.
In general, $a_n\not\to0$ as $n\to\infty$. Indeed, let \begin{equation*} q_n:=\frac{\mu_n}{\lambda_n+\mu_n}. \end{equation*} Then, by (1), for all natural $n$ $$a_n\ge q_na_{n-1}\ge q_n q_{n-1}a_{n-2}\ge\cdots\ge q_n q_{n-1}\cdots q_1\ge Q:=\prod_{j=1}^\infty q_j>0 $$ if $\sum_{j=1}^\infty(1-q_j)<\infty$. So, in such a case we indeed have $a_n\not\to0$ as $n\to\infty$.
I guess you misunderstood something in Karlin's book. It could help if you reproduced here all the relevant definitions and exact statements from that book or, at least, gave us an accessible reference to the book.