Converging Sequences and Limits: Help with Induction Proof?

[Swamifez]

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)/2a) 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.

 

[quasar987]

1° show it's true for n=1 first.

2° assume it's true for n=m-1, i.e. assume a_{m-1}&lt;a_{m}&lt;b_{m}&lt;b_{m-1}

3° Use the part of the induction hypothesis that say a_{m}&lt;b_{m} to prove a_{m+1}&lt;a_m and b_{m+1}&gt;b_m. For the part a_{m+1}&lt;b_{m+1}, notice that since the a_i and b_i are positive, it is equivalent to showing that (a_{m+1})^2&lt;(b_{m+1})^2, i.e. that a_nb_n&lt;\frac{a_n^2+b_n^2}{4}+\frac{a_nb_n}{2}, etc. (think perfect square). That's enough hints. Go think for another hour.