## anonymous one year ago sequence {X_n } is defined by X(n+1) = 2X(n)-X^2(n). If 0<X(0)<1 so is 0<X(n)<1 that also for all integers n>0

1. anonymous

$X _{n}$ is defined by $X _{n+1}=2X _{n}-X ^{2}_{n}$ show that if $0<X _{0}<1$ so is $0<X _{n}<1$ for all integers n>0

2. anonymous

@IrishBoy123 wanna give me some hints with this one? i have tried using the same method..

3. IrishBoy123

try something like $$X_n = 2 - \frac{X_{n+1}}{X_{n}}$$ as $$0<X_n <1$$ then $$0< 2 - \frac{X_{n+1}}{X_{n}}<1$$ do each inequality separately see if it leads somewhere...

4. anonymous

Something else you can try: Let $$f(x)=2x-x^2$$. Then $$f'(x)=2-2x$$. For $$0<x<1$$, you have that $$f'(x)>0$$, which would suggest that $$\{X_n\}$$ is an increasing sequence (at least if $$0<X_n<1$$). This smells like an induction proof waiting to happen.