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Suppose that $0 < a_n < 1, \sum_{n=1}^\infty a_n=\infty,b_0=1, b_{n+1}=(1-a_n)b_n$. Find $\lim_{n\to \infty} b_n$.

It seems $\lim_{n\to \infty} b_n=0$. Since $b_{n+1}=(1-a_n)b_n < b_n$, $b_n \geq 0$ is strictly decreasing and bounded below,thus converges. Furthermore, if $\inf b_n=0$, then $\lim_{n\to \infty} b_n=0$. Perhaps $\sum_{n=1}^\infty a_n=\infty$ can be used for this purpose or there is a better proof, but how?

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1 Answer 1

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Take logs, and show $\lim\ln b_n=-\infty$.

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