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 $a_{m-1}<a_{m}<b_{m}<b_{m-1}$

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