1. ## Induction

Show by induction:

$\displaystyle x_{n+2}=\frac{x_{n}+x_{n+1}}{2}$

$\displaystyle x_{1}=0, x_{2}=1,$

that $\displaystyle 0\leq x_{n}\leq 1$$\displaystyle , n= 1,2,3,...$

If i suppose that $\displaystyle 0\leq x_{k+2}=\frac{x_{k}+x_{k+1}}{2}\leq 1$ is valid,
then i want to show that $\displaystyle 0\leq x_{k+3}=\frac{x_{k+1}+x_{k+2}}{2}\leq1$ is valid by using my assumption. But i can't find a way to do this.
Maybe this is totally wrong way to do it.

/regards

2. ## Re: Induction

Don't forget to show the result for n=1 and n=2.
Since $\displaystyle 0\leq x_{k+1}\leq 1$ and $\displaystyle 0\leq x_{k+2}\leq 1$ then $\displaystyle 0\leq x_{k+1}+x_{k+2}\leq 2$. Intuitively, the result is quite clear: if we take two points between 0 and 1, their middle will be between 0 and 1.