What does $(a_n)_n \in A^{\mathbb{N}}$ mean?

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What does $(a_n)_n \in A^{\mathbb{N}}$ mean?

What kind of sequence is that? How does the indexing work?

What's the $A$ to natural numbers power?

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That power notation indicates the set of functions, $A^\mathbb{N} = \{f: \mathbb{N} \to A\}$, which in this case means the set of infinite sequences of elements of $A$.