gnoka wrote: Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below: $ a_{0}=1 $ $a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$ Prove $a_{2019} < \frac{1}{2} < a_{2018}$ I think we just need to prove a more generalized problem: Define sequence {$a_{n}$} as below: $a_{0}=1$ $a_{n+1}=a_{n}-\frac{a_{n}^{2}}{k+1}$, $n=0,1,2, \ldots, 2018$ Prove $a_{k+1} < \frac{1}{2} < a_{k}$

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