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My questions are about a sequence or function with several variables.

  • I vaguely remember some while ago one of my teachers said taking limits of a sequence or function with respect to different variables is not exchangeable everywhere, i.e. $ \lim_n \lim_m a_{n,m} \neq \lim_m \lim_n a_{n,m}, \quad \lim_x \lim_y f(x,y) \neq \lim_y \lim_x f(x,y).$ So my question is what are the cases or examples when one can exchange the order of taking limits and when one cannot, to your knowledge? I would like to collect the cases together, and be aware of their difference and avoid making mistakes. If you could provide some general guidelines, that will be even nicer!
  • To give you an example of what I am asking about, this is a question that confuses me: Assume $f: [0, \infty) \rightarrow (0, \infty)$ is a function, satisfying $ \int_0^{\infty} x f(x) \, dx < \infty. $ Determine the convergence of this series $\sum_{n=1}^{\infty} \int_n^{\infty} f(x) dx$.

    The answer I saw is to exchange the order of $\sum_{n=1}^{\infty}$ and $\int_n^{\infty}$ as follows: $ \sum_{n=1}^{\infty} \int_n^{\infty} f(x) dx = \int_1^{\infty} \sum_{n=1}^{\lfloor x \rfloor} f(n) dx \leq \int_1^{\infty} \lfloor x \rfloor f(x) dx $ where $\lfloor x \rfloor$ is the greatest integer less than $x$. In this way, the answer proves the series converges. I was wondering why the two steps are valid? Is there some special meaning of the first equality? Because it looks similar to the tail sum formula for expectation of a random variable $X$ with possible values $\{ 0,1,2,...,n\}$: $\sum_{i=0}^n i P(X=i) = \sum_{i=0}^n P(X\geq i).$ The formula is from Page 171 of Probability by Jim Pitman, 1993. Are they really related?

Really appreciate your help!

  • 0
    Possible repeat of http://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits2011-01-19

3 Answers 3

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A simple example of a doubly indexed sequence $a_{m,n}$ for which you cannot exchange limits is given in Rudin's "Principles..." Example 7.2 pg. 144:

Let $a_{m,n} = \frac{m}{m+n}$, then $\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}a_{m,n} = 1,$ but $\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}a_{m,n} = 0.$

Here is a previous post on this question which seems to thoroughly answer your other questions: When can you switch the order of limits?

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A simpler example where one cannot switch the limits: $a_{m,n}=\begin{cases}0&\text{if }m\le n\\1&\text{if }m>n\end{cases}$

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$f(x,y)=\frac{x-y}{x+y}.$

You can also check the limit at the point $(0,0)$. Typical example of calculus.