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Help with Induction proof

  1. Dec 20, 2006 #1
    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.
     
    Last edited: Dec 20, 2006
  2. jcsd
  3. Dec 20, 2006 #2

    quasar987

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    1° show it's true for n=1 first.

    2° assume it's true for n=m-1, i.e. assume [itex]a_{m-1}<a_{m}<b_{m}<b_{m-1}[/itex]

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