# Fubini theorem for sequences

I want to find a counter example

This is the Fubini theorem for sequences:

If $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty,$$

then

$$\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}a_{mn}=\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}a_{mn}.$$

Then, does there exist a sequence $\{a_{mn}\}$ such that the left and the right hand sides of equality are finite but are not equal?

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Fubini's theorem (in the special case of sequences) says that if $\sum\sum|a_{mn}|<\infty$ then interchanging the order of summation in $\sum_m \sum_n a_{mn}$ is acceptable. Note the lack of modulus signs here. In contrast, interchanging the order of summation is always acceptable in $\sum\sum |a_{mn}|$, since the summands are nonnegative (this is sometimes called Tonelli's theorem). In this case, equality means that if either side is finite then they both are and they're equal.
If $\sum\sum|a_{mn}|=\infty$, then it may happen that $\sum_m \sum_n a_{mn}$ and $\sum_n \sum_m a_{mn}$ are different, even if they are both finite. Indeed, try summing the rows and columns of the following infinite matrix:
$$\left(\begin{array}{cccccc} 1 & - 1 & 0 & 0 & 0 &\cdots\\ 0 & 1 & -1 & 0 & 0 &\cdots\\ 0 & 0 & 1 & -1 & 0 &\cdots\\ 0 & 0 & 0 & 1 & -1 &\cdots\\ 0 & 0 & 0 & 0 & 1 &\cdots\\ \vdots & \vdots & \vdots &\vdots &\vdots & \ddots\\ \end{array}\right)$$
Yes, but this doesn't satisfy the condition $\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty$, does it? – Old John Sep 10 '12 at 14:37
I added the absolute values in the second formula of the question, given that the OP missed it at all (he simply write $\sum_m\sum_n=\sum_n\sum_m$). – enzotib Sep 10 '12 at 14:55