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Order of limits

  1. Jun 15, 2014 #1
    In the sense most ample and general of limits, the following identitie is true:
    $$\\ \lim_A \lim_B = \lim_B \lim_A$$
    ?
     
    Last edited: Jun 15, 2014
  2. jcsd
  3. Jun 15, 2014 #2

    micromass

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    No, it's not.
     
  4. Jun 15, 2014 #3

    jbunniii

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    Simple counterexample: consider the function ##x : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## defined by
    $$x(m,n) =
    \begin{cases}
    1 & \text{if }m > n \\
    0 & \text{otherwise}
    \end{cases}$$
    For every ##m##, we have ##\lim_{n \rightarrow \infty}x(m,n) = 0## and therefore ##\lim_{m \rightarrow \infty}\lim_{n \rightarrow \infty}x(m,n) = 0##.

    Similarly, for every ##n##, we have ##\lim_{m \rightarrow \infty}x(m,n) = 1##, and therefore ##\lim_{n \rightarrow \infty} \lim_{m \rightarrow \infty}x(m,n) = 1##.
     
  5. Jun 16, 2014 #4
    and exist general cases where ##\\ \lim_A \lim_B = \lim_B \lim_A## is true?
     
  6. Jun 16, 2014 #5

    micromass

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    Yes, and that's actually what a giant part of real analysis is about: finding when you can switch two limits.

    Please see Knapp's "Basic Real Analysis". In the first chapter he already gives ##2## general situations where it's true.
    Aside from that, there are many specialized situations where it is also true, these are incredibly important theorems. A small selection:
    http://en.wikipedia.org/wiki/Monotone_convergence_theorem#Lebesgue.27s_monotone_convergence_theorem
    http://en.wikipedia.org/wiki/Dominated_convergence_theorem
    http://en.wikipedia.org/wiki/Fubini's_theorem
    http://en.wikipedia.org/wiki/Power_series#Differentiation_and_integration
     
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