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Question about liminf of the sum of two sequences

  1. Oct 22, 2009 #1
    I know that for any two real sequences x_n and y_n, we have

    [tex]
    \liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).
    [/tex]

    I also know that, if one of the sequences converges, the inequality becomes equality. My question is this: If I've managed to show that

    [tex]
    \liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n = \liminf_{n\to \infty}(x_n + y_n),
    [/tex]

    can I conclude that one, or both, of the sequences converge? A simple yes/no would suffice, but (of course) I'd prefer a short proof or counterexample. Thanks!
     
  2. jcsd
  3. Oct 22, 2009 #2

    statdad

    User Avatar
    Homework Helper

    What if

    [tex]
    x_n = y_n = (-1)^n
    [/tex]
     
  4. Oct 22, 2009 #3
    Lame! I was hoping both of the sequences had to converge! And what a simple counterexample to prove me wrong! Thanks, though :smile:
     
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