I'm working through exercises in Axler's "Linear Algebra Done Right" and trying to verify Example 1.35 (e) on page 19, which is stated as follows: 

*(e) The set of all sequence of complex numbers with limit 0 is a subspace of $\mathbb{C}^{\infty}$.*

**My main question is: Wouldn't the set of all sequences of complex numbers for example include the sequence $(1,2,3)$ which is to my understanding not in $\mathbb{C}^{\infty}$ thereby making this not a subset and therefore not a subspace?**

My understanding is that $\mathbb{C}^\infty$ contains all lists of length $\infty$ of complex numbers just as $\mathbb{R}^3$ contains all lists of length 3 of real numbers.

Also I now understand a list, tuple, sequence and vector to all be names for ordered collections of objects (generally numbers) of a certain length, and I'm wondering whether this is correct? Before this example I hadn't really made a connection between vectors and sequences, but it seems there is no difference in definition. 

There is a set of solutions available at <http://linearalgebras.com/1c.html> (question 2e in this case) which doesn't address my question.