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!
Best Answer
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?