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If {$a_n$} is a sequence of natural numbers, can we write {$a_n$} $\in\mathbb N$ ? Thanks.

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That's abuse of notation. Sometimes, a sequence is written as $$\{a_n\}_{n\in\mathbb N}\subset \mathbb N$$ That too is abuse of notation, but perhaps a little clearer. –  Thomas Andrews Feb 15 '13 at 15:38
    
Maybe something like $\{a_n\}\in \mathbb{N}^\mathbb{N}$ should be possible –  Dominic Michaelis Feb 15 '13 at 15:39

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No. A sequence of natural numbers is a function from $\Bbb N$ to $\Bbb N$. The notation $\{a_n\}\in\Bbb N$, says that the object $\{a_n\}$ is a natural number. In fact it’s a set with one element, that being a particular term of the sequence, or a sloppy abbreviation for $\{a_n:n\in\Bbb N\}$, in which case it’s a set of natural numbers. In neither case is it a natural number or a sequence of natural numbers.

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