I recently encountered the following definition of a finite-dimensional vector space in Axler's Linear Algebra Done Right: A vector space is called finite-dimensional if some list of vectors in it spans the space.
This I was then thinking about $V = \{(x, 0, 0, \cdots): x \in \mathbb{R} \}$ i.e., the set of all infinite sequences whose first value is some real number and for which all other values are 0.
The book defines ${\mathbb{R}}^{\infty} = \{(x_1, x_2, \cdots): x_j \in \mathbb{R} \text{ for } j = 1, 2, \cdots \}.$ He defines addition and scalar multiplication with ${\mathbb{R}}^{\infty}$ as you'd expect: $(x_1, x_2, \cdots) + (y_1, y_2, \cdots) = (x_1+y_1, x_2+y_2, \cdots)$ and $\lambda(x_1, x_2, \cdots) = (\lambda x_1, \lambda x_2, \cdots)$.
Clearly $V$ is a subspace of $\mathbb{R}^{\infty}$. Now since any $v = (x, 0, 0 \cdots) \in V$ can be written $v = x(1, 0, 0, \cdots)$ where $x \in \mathbb{R}$ and $(1, 0, 0, \cdots) \in V$, $(1, 0, 0, \cdots)$ alone spans $V$. Since this list of just one vector in $V$ spans $V$, I am inclined to think that $V$ is finite-dimensional.
This does not mesh with any pre-existing intuition I had about what finite-dimensional vector spaces were, nor can I find any examples like this of a finite-dimensional vector space online, which leads me to think there is an error in my reasoning. Is this correct? If not, can someone point me to the error in my reasoning?