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.
The word "sequence" can sometimes refer to finite sequences (or indeed sequences with even more general index sets), but in many contexts it is used to refer only to infinite sequences (with index set $\mathbb{N}$). That is clearly the intended meaning in the context of this problem. So, for the purposes of this problem, a "sequence of complex numbers" is an infinite list of complex numbers.