Again! sequences n limits?

Question

x_1 and y_1 are given, x_1>y_1>0

x_n+1 = (x_n+y_n)/2
y_n+1 = sqrt(x_n*y_n)

a)proof y_n< y_n+1
b)proof y_1 < x_n+1 < x_n

c)proof 0<(x_n+1) - (y_n+1)< (x_1 - y_1)2^n

d)explain why {x_n} and {y_n} are convergent and have the same limit


anything will help
Thank you all of u guys

Answer 1

a) for n=2 : x2 x2>y1
y2 y2>y1

this so all because of x_1>y_1 and using the definitions of x_n,y_n

from this it is straight forward with induction to proof a,b,c