Formal logic syntax for all sequence members in a set

Question

How do I write the formal expression for 'All members of the sequence A are in the set B'?

I have seen ∀h.h∈N means 'All h are in the set of Natural numbers' But can't find anything on how to apply this to sequence members.

Answer 1

To restrict a universal to a domain, we may use either of: $~\forall c\in A ~. c\in B~$ or $~\forall c~.(c\in A\to c\in B)$.

Also, as others have stated, as a sequence $A$ itself may be defined as a function mapping the naturals to some set, so $\forall n\in\Bbb N~.A(n)\in B$ is another posibility.

Answer 2

By a sequence we usually mean a function $a : \mathbb{N} \to X$ where $X$ is some set. To formally express the statement 'all terms of the sequence $a$ are in the set $B$', I would write

$$\{ a_n : n \in \mathbb{N} \} \subseteq B$$

or

$$(\forall n \in \mathbb{N}) \, a_n \in B$$

(I think the first one is a bit more readable.)