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Homework Help: Limit of a sequence

  1. Jun 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that the limit of the sequence {Sqrt(n+1)-Sqrt(n)} = 0.

    2. Relevant equations


    3. The attempt at a solution

    I know that I must multiply by the conjugate to come up with 1/(Sqrt[n+1]-Sqrt[n]) and that the limit of this is clearly 0. I am having trouble solving this equation in terms of epsilon.
     
  2. jcsd
  3. Jun 8, 2010 #2
    Be careful of the sign; multiplying by the conjugate gives you 1/{sqrt(n+1) + sqrt(n)}. For all ε > 0, there exists 1/m < ε for some positive integer m. (Why?) How should you choose N so that 1/{sqrt(n+1) + sqrt(n)} < 1/m < ε whenever n > N?
     
  4. Jun 8, 2010 #3
    Ah, yes, the sign. Thank you! Now... To choose N, can I ignore the sqrt(n) part of the denominator since sqrt(n+1) > sqrt(n) so, being in the denomominator, the number is smaller? So could I let N = (1/epsilon^2) - 1?
     
  5. Jun 8, 2010 #4
    That should work fine; by convention we let N be an integer, so you can set N to be the ceiling of what you have.
     
  6. Jun 8, 2010 #5
    Thank you!
     
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