How far along the 'path' have you gotten?
Have you been able to start with the basic step?
Please tell us where you are in the proof.
I've been working on this for the past hour, but haven't gone anywhere with it. If anyone can help to complete it, it would be highly appreciated. Thanks
Let 0< a1< b1 and define
an+1= √anbn
bn+1=(an+bn)/2
a) Use induction to show that
an<an+1<bn+1<bn
Thus prove that an and bn converge.
b) Prove that they have the same limit.
Suppose that for some that:
then as ,
and as .
Also by the geometric-arithmetic mean inequality:
(equality is not a possibility here as ),
so we have shown that:
.
The base case is demonstrated by showing that:
using essentialy the same methods, which proves the required result by
mathematical induction.
RonL