$n\in N, a_{n+1} = a_n (1 - a_n)$ , $0 < a_0 < 1$. Prove that $\lim\limits_{n\to\infty} a_n\cdot n = 1$.

sequences-and-series calculus limits

Solution 1:

Prove that $\frac{1}{a_{n+1}}-\frac{1}{a_n} \rightarrow 1$.

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