Putnam 2011 A2

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for $n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove that $$S = \sum_{n=1}^\infty \frac{1}{a_1...a_n}$$ converges, and evaluate $S$.