_{1}< x

_{1}and set

x

_{n+1}= (1/2)(x

_{n}+ y

_{n}) and y

_{n+1}= sqrt(x

_{n}y

_{n})

Prove that 0 < x

_{n+1}- y

_{n+1}< (x

_{1}- y

_{1})/2

^{n}for n in

**N**

Provided solution:

x

_{n+1}- y

_{n+1 }= (1/2)(x

_{n}+ y

_{n}) - sqrt(x

_{n}y

_{n}) < (1/2)(x

_{n}+ y

_{n}) - y = (1/2)(x

_{n}- y

_{n})

Hence by induction and by the fact that 0 < y

_{n}< x

_{n}for n in

**N,**0 < x

_{n+1}- y

_{n+1}< (x

_{1}- y

_{1})/2

^{n }

What I do not understand is the last part of this explanation. I understand that, by induction, x

_{n+1}- y

_{n+1}< (x

_{1}- y

_{1})/2, but why 2

^{n}?

Thanks in advance.