Identifying a sequence as subset of subspace If I have some sequence $\mathcal A = (a_i)$ of objects $a_i$ (maybe finite, maybe countably infinite) how can I say that those objects all exist in some subspace $S$?  Is it correct to say $\mathcal A \subseteq S$?  I'm not sure because $\mathcal A$ isn't a set it's a sequence.

You are correct in your assertion that $(a_i)\subseteq S$ lacks some formality, but I think it would be understood as intended if you wrote it that way.
Here are some other options:


*

*Use $(a_i) \in S^{\mathbb{N}}$ if you have a sequence in the usual sense.

*Use $(a_i) \in S^N$ if you have a sequence of finite length.

*Introduce $\{a_i\}\subseteq S$ and use the object $(a_i)$ when needed.

