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?
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?
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.