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Question Number 219414 by universe last updated on 24/Apr/25

$$\mathrm{given}\mathrm{the}\mathrm{recursive}\left\{{\mathrm{a}}_{\mathrm{n}}\right\}\mathrm{define}\mathrm{by}\mathrm{setting}$$$${\mathrm{a}}_1\in (0,1),{\mathrm{a}}_{\mathrm{n}+1}={\mathrm{a}}_{\mathrm{n}}(1-{\mathrm{a}}_{\mathrm{n}}),\mathrm{n}⩾1$$$$\mathrm{prove}\mathrm{that}\left(1\right)\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }{\mathrm{na}}_{\mathrm{n}}=1$$$$\left(2\right){\mathrm{b}}_{\mathrm{n}}=\mathrm{n}(1-{\mathrm{na}}_{\mathrm{n}})\mathrm{is}\mathrm{a}\mathrm{incresing}\mathrm{sequence}$$$$\mathrm{and}\mathrm{diverge}\mathrm{to}\mathrm{\infty }$$$$\left(3\right)\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{\mathrm{n}(1-{\mathrm{na}}_{\mathrm{n}})}{\ln \left(\mathrm{n}\right)}=1$$

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