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 $n$. 

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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 $n\to\infty$. 

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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.