Having trouble with induction proof and limit

Can someone just show me a full blown proof of this problem, I really would have posted everything I had, but its really hard for me to type everything I had. 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.

This is what I got so far

1) show it's true for n=1 first.

I derived these an+1=square root of anbn>square root of anan =an

an<an+bn/2<bnanan=an => an<bn+1<bn

I couldn't derive anything for the second term