Question Number 86326 by ~blr237~ last updated on 28/Mar/20
![Let f a continue function acknowleding α as a fix point on [0,1].F a function such as (dF/dx)=f(x) ∀ n , u_(n+1) =((F(u_n )−F(α))/(u_n −α)) Prove that lim_(n→∞) u_n =α](Q86326.png)
$${Let}\:{f}\:{a}\:{continue}\:{function}\:{acknowleding} \\ $$$$\alpha\:{as}\:{a}\:{fix}\:{point}\:{on}\:\left[\mathrm{0},\mathrm{1}\right].{F}\:\:{a}\:{function}\:{such}\:{as}\:\frac{{dF}}{{dx}}={f}\left({x}\right) \\ $$$$\forall\:{n}\:,\:\:{u}_{{n}+\mathrm{1}} =\frac{{F}\left({u}_{{n}} \right)−{F}\left(\alpha\right)}{{u}_{{n}} −\alpha}\: \\ $$$${Prove}\:{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{u}_{{n}} \:=\alpha \\ $$