Order of limits

1. Jun 15, 2014

Jhenrique

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. Jun 15, 2014

micromass

Staff Emeritus
No, it's not.

3. Jun 15, 2014

jbunniii

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 $\lim_{n \rightarrow \infty} \lim_{m \rightarrow \infty}x(m,n) = 1$.

4. Jun 16, 2014

Jhenrique

and exist general cases where $\\ \lim_A \lim_B = \lim_B \lim_A$ is true?

5. Jun 16, 2014

micromass

Staff Emeritus
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