Affine maps and affine subspaces

Question

In the Appendix of John M Lee's Textbook "Introduction to Smooth Manifolds" there is the following Exercise B.17

Suppose $V$ is a finite-dimensional vector space. Show that every affine subspace of $V$ is of the form $F^{-1}(z)$ for some affine map $F:V \to W$ and some $z\in W$.

Some definitions:

1. An affine subspace of $V$ is any subset of the form $v+S=\{v+w:w\in S\}$ with $v\in V$ and $S$ linear subspace of $V$.
2. If $V$ and $W$ are vector spaces, a map $F:V\to W$ is called an affine map if it can be written in the form $F(v)=w+Tv$ for some linear map $T:V\to W$ and some fixed $w\in W$.

Here is my argument

Let $v_0+S$ denote the generic affine subspace of $V$. Let $\pi:V\to V/S$ denote the canonical projection. Then $\pi$ is an affine map and $\pi^{-1}(v_0+S)=v_0+S$. So the thesis is true with $W=V/S, F=\pi, z=v_0+S$.

It seems correct to me but I don't use the hypothesis that $V$ is finite dimensional. So have I misunderstood the statement of the exercise or the proof is not correct?