Putnam 1976 B3

B-3. Suppose that we have $n$ events $A_{1}, \ldots, A_{n}$, each of which has probability at least $1-a$ of occurring, where $a<1 / 4$. Further suppose that $A_{1}$ and $A$, are mutually independent if $|i-j|>1$, although $A_{1}$ and $A_{i+1}$ may be dependent. Assume as known that the recurrence $u_{k+1}=u_{k}-a u_{k-1}, u_{0}=1, u_{1}=1-a$, defines positive real numbers $u_{k}$ for $k=0,1, \ldots$. Show that the probability of all of $A_{1}, \ldots, A_{n}$ occurring is at least $u_{n}$.