Question 139
Let $f: \mathbf{R} \rightarrow (0,+\infty)$ be a differentiable, strictly decreasing function with $f(0)=1$ and $f^{\prime}(0)=-1$. Let the sequence $\left(a_{n}\right)_{n=1}^{\infty}$ be defined by $a_{1}=1$ and $a_{n+1}=a_{n} f\left(a_{n}\right), \forall n \geq 1$. Prove that the limit $\lim _{n \rightarrow \infty}\left(n a_{n}\right)$ exists and compute this limit.