Pick a basis \(e_1, \dots, e_n\) of \(V_0\). Extend each \(\rho(e_i)\) in the form

\[ \rho(e_i) \;=\; \sum_{j=1}^{N} f_{ij} \otimes a_{ij}, \]

where \(\{f_{ij}\}_{i,j}\) is a finite subset of \(V\) and \(\{a_{ij}\}_{i,j}\) is a finite subset of \(A\) (such an expansion exists because \(V \otimes_k A\) is spanned by elementary tensors and \(\rho(e_i)\) has only finitely many nonzero terms in any chosen basis of \(A\)). Let

\[ W \;=\; \operatorname{span}_k\bigl(\, \{e_1, \dots, e_n\} \cup \{f_{ij}\}_{i,j} \,\bigr) \;\subseteq\; V. \]

Then \(W\) is finite-dimensional and contains \(V_0\). We check that \(W\) is a sub-comodule, i.e.\ \(\rho(W) \subseteq W \otimes_k A\).

For \(\rho(e_i)\) this holds by construction: \(\rho(e_i) = \sum f_{ij} \otimes a_{ij}\) and each \(f_{ij} \in W\), so \(\rho(e_i) \in W \otimes A\).

For \(\rho(f_{ij})\), apply coassociativity:

\[ (\operatorname{id}_V \otimes \Delta) \circ \rho \;=\; (\rho \otimes \operatorname{id}_A) \circ \rho. \]

Applied to \(e_i\), the left side gives \(\sum_j f_{ij} \otimes \Delta(a_{ij})\) and the right side gives \(\sum_j \rho(f_{ij}) \otimes a_{ij}\). The linear independence of \(\{a_{ij}\}_j\) (after refining the family to a linearly independent subset, replacing the \(f_{ij}\) by linear combinations) implies that each \(\rho(f_{ij})\) is a linear combination of elements \(f_{i\ell} \otimes \text{(coefficient in } A\text{)}\), hence lies in \(W \otimes A\). Concluding, \(W\) is a sub-comodule.
\(\square\)

Corollary. Let \(A\) be a finitely generated commutative Hopf \(k\)-algebra and let \(a_1, \dots, a_m \in A\) be a finite set of \(k\)-algebra generators. There exists a finite-dimensional sub-comodule \(W \subseteq A\) of the regular representation containing each \(a_i\); in particular, \(W\) generates \(A\) as a \(k\)-algebra.

Proof of corollary. Apply the lemma to \(V = A\), \(\rho = \Delta\), and \(V_0 = \operatorname{span}_k\{a_1, \dots, a_m\}\).
\(\square\)