Step 1: replace \(G\) with a Hopf algebra. By the essential-image characterization of Spec, \(G\) is isomorphic to \(\operatorname{Spec} A\) for some commutative Hopf \(k\)-algebra \(A\). Locally of finite type plus affine plus base \(\operatorname{Spec} k\) implies \(A\) is finitely generated as a \(k\)-algebra.

Step 2: find an algebra-generating sub-comodule of the regular representation. Choose a finite set \(a_1, \dots, a_m\) of \(k\)-algebra generators of \(A\). By the corollary to the finite-generation lemma, there is 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.

Step 3: convert the comodule into a representation. By the equivalence of representations and comodules, the comodule structure \(\Delta|_W : W \to W \otimes_k A\) corresponds to a morphism of group schemes

\[ \rho : G \;\to\; \operatorname{GL}(W). \]

Step 4: \(\rho\) is a closed immersion. The morphism \(\rho\) corresponds, on coordinate Hopf algebras, to a \(k\)-algebra map

\[ \rho^* : \mathcal{O}(\operatorname{GL}(W)) \;\to\; A. \]

The coordinate ring \(\mathcal{O}(\operatorname{GL}(W))\) is generated as a \(k\)-algebra by the matrix coefficient functions \(c_{ij}\) of the standard representation of \(\operatorname{GL}(W)\) (together with \(\det^{-1}\)). The image \(\rho^*(c_{ij})\) is the matrix coefficient of the representation \(\rho\) on \(W\): writing the coaction \(\Delta|_W : W \to W \otimes_k A\) in a basis \(e_1, \dots, e_n\) of \(W\) as \(\Delta(e_i) = \sum_j e_j \otimes b_{ij}\), one has \(\rho^*(c_{ij}) = b_{ij}\).

We claim the \(b_{ij}\) generate \(A\) as a \(k\)-algebra. For any \(w \in W\), write \(\Delta(w) = \sum w_\alpha \otimes b_\alpha\) with \(w_\alpha \in W\) and \(b_\alpha\) in the \(k\)-subalgebra generated by the \(b_{ij}\). Applying \(\varepsilon\) to the first tensor factor and using the counit identity gives \(w = \sum \varepsilon(w_\alpha) b_\alpha\), so \(w\) lies in the \(k\)-subalgebra generated by the \(b_{ij}\). Since \(W\) generates \(A\) as a \(k\)-algebra and \(W\) is contained in this subalgebra, the \(b_{ij}\) generate \(A\).

Hence the image of \(\rho^*\) contains a \(k\)-algebra generating set of \(A\), so \(\rho^*\) is surjective. A surjective ring homomorphism corresponds to a closed immersion of affine schemes. Therefore \(\rho\) is a closed immersion.

Step 5: conclude. The pair \((W, \rho)\) is a faithful finite-dimensional representation of \(G\).
\(\square\)