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Homework Help: Real analysis: Sequences question

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data
    If Xn is bounded by 2, and [tex]|X_{n+2} - X_{n+1}| \leq \frac{|X^2_{n+1} - X^2_n|}{8} [/tex], prove that Xn is a convergent sequence.

    2. Relevant equations

    3. The attempt at a solution
    I believe the solution lies in proving Xn a Cauchy sequence, but I'm not sure how to work it out. I considered |Xn - Xm| adding and subtracting the terms between m and n but i got stuck.
    I also tried to check for telescoping, with no luck.
    Last edited: Apr 13, 2009
  2. jcsd
  3. Apr 13, 2009 #2
    Factor [tex]X^2_{n+1} - X^2_n[/tex].

    Then get an upper bound for [tex]\frac{|X^2_{n+1} - X^2_n|}{8}[/tex].
  4. Apr 13, 2009 #3
    \frac{|X^2_{n+1} - X^2_n|}{8} = \frac{|(X_{n+1} + X_n)(X_{n+1} - X_n)|}{8} \leq \frac{|X_{n+1} - X_n|}{2}

    Iterating, I reached [tex] |X_{n+2} - X_ {n+1}| \leq \frac{|X_2 - X_1|}{2^n} [/tex]
    I'm not sure if I continued in the right track.. but I'm stuck.
    Thanks for your input.
  5. Apr 13, 2009 #4


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    Now go back to trying to show the sequence is Cauchy by adding and subtracting terms in |xn-xm|. Use the triangle inequality.
  6. Apr 13, 2009 #5
    Or could even look at the sequence of partial sums, getting it bounded above.
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