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Another sequence convergence proof

  1. Sep 27, 2007 #1
    1. The problem statement, all variables and given/known data

    Let [tex]y_n := \sqrt{n+1} - \sqrt{n}[/tex] for [tex]n \in \mathbb{N}[/tex]. Show that [tex] (y_n)[/tex] converges.

    2. Relevant equations

    3. The attempt at a solution

    I see that it converges to 0. I just need a nudge in the right direction at getting into [tex]| \sqrt{n+1} - \sqrt{n} - 0 | = | \sqrt{n+1} - \sqrt{n} |[/tex] to show it's less than any [tex]\epsilon > 0[/tex]. Any manipulating I've tried so far makes the terms way too big to work with.
     
    Last edited: Sep 27, 2007
  2. jcsd
  3. Sep 27, 2007 #2

    morphism

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    How about

    [tex]\left(\sqrt{n+1} - \sqrt{n}\right) \cdot \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}[/tex]
     
  4. Sep 27, 2007 #3
    ohhhh, i see it now.
     
  5. Jan 21, 2009 #4
    What do you do after
    1/(sqrt{n+1)+sqrt{n}) ??
     
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