A possible typo in a proof about closure in textbook Introduction to Set Theory 3rd by Hrbacek and Jech

Question

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech introduces the below theorem as well as its proof:

In the proof, there is a statement which I highlight in red color:

If $\langle a_n \mid n \in \Bbb N \rangle$ is a sequence in $D$, then $\langle a_n \mid n \in \Bbb N \rangle \in C_\alpha$ for some $\alpha < \omega_1$

I would like to ask two questions regarding this statement:

1. Is it correct that "$\langle a_n \mid n \in \Bbb N \rangle$ is a sequence in $D$" means $\forall n \in \Bbb N:a_n \in D$?

2. I think $\langle a_n \mid n \in \Bbb N \rangle \in C_\alpha$ is a typo. Instead, it should be $\forall n \in \Bbb N:a_n \in C_\alpha$. Please confirm if my understanding is correct!

Thank you for your help!

Answer 1

Your interpretations are correct. However, I'm not sure I'd call this a typo: notation like "$(a_i)_{i\in I}\in X$" (and language like "a sequence in $X$") is a very common abuse of notation for "$\forall i\in I(a_i\in X)$." So I think Jech wrote what he wanted to, although it is slightly incorrect.