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Question Number 209342 by essaad last updated on 07/Jul/24

Answered by Berbere last updated on 07/Jul/24

$$U_{n+1}=f\left(U_n\right)$$$$x\stackrel{f}{\to }\frac{x}{2}+\frac{x^2}{4};fincrease$$$$f\left([0,1]\right)=[0,\frac{3}{4}]\subset [0,1]$$$$\Rightarrow U_n\in [0,1]$$$$U_{n+1}-U_n=-\frac{U_n}{2}+\frac{U_n^2}{4}=\frac{U_n(U_n-2)}{4}<0$$$$U_{n}decreaseBoundedCvtol\in [0,1]$$$$sincefiscontinusin[0,1]\Rightarrow l=f\left(l\right)\iff l=\frac{l}{2}+\frac{l^2}{4}$$$$\Rightarrow \frac{l}{2}=\frac{l^2}{4}\Rightarrow l(2-l)=0\Rightarrow l=0$$

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