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This is probably a trivial question, but what is the difference between those two?

$\bigcap_{n=1}^\infty = \{x \mid \forall n \in \mathbb N, x \in A_n\}$

What does the other intersection mean?

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    That's what I thought at first, but it was written that way in two separate problems. So does the notation on the right have any meaning at all, or is it actually meaningless?2012-10-25

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Usually we define merely $\bigcap_{i\in I}A_i=\{x\mid \forall i\in I\colon x\in A_i\}$ where $I$ is some nonempty index set. Alternate notations are common for two special cases:

(i) If $I=\{n, n+1, \ldots, m\}$, we write $\bigcup_{i=n}^m A_i$ for $\bigcup_{i\in I}A_i$.

(ii) If $I=\{i\in\mathbb N\mid i\ge n\}$, we write $\bigcup_{i=n}^\infty A_i$ for $\bigcup_{i\in I}A_i$.

The notation you exhibit has never occured to me. It doesn't match notations for e.g. sums with $\Sigma$ either.

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    Ok, thank you. I'll assume that it was a typo until I can ask my professor about it tomorrow. That certainly makes the homework problem I am trying to do much easier.2012-10-25