Let $C$ be a coalgebra over a field and $M$ be a $C$-comodule. Then it's well-known that every element of $M$ is contained in a finite-dimensional subcomodule $M' \subset M$. This is for example an adaptation of proof given in Hovey's Model categories (Lemma 2.5.1, with notations changed a bit):
Let $\{c_i\}$ be a basis of $C$ and write the coaction $\gamma : M \to C \otimes M$ as: $$\gamma(m) = \sum_i c_i \otimes m_i,$$ where only a finite number of $m_i$ may be nonzero. For a fixed $m \in M$, let $M'$ be the finite-dimensional subspace of $M$ spanned by the nonzero $m_i$. By counitality, $M'$ contains $m$, and using coassociativity of $\gamma$, $M'$ is a $C$-comodule.
As you can see, this proof requires choosing a basis of $C$. Is it possible to rewrite the proof to avoid choosing this basis? If not, is it possible that the result becomes false in a setting where choosing a basis is not always possible (infinite dimensional $C$ and no axiom of choice)?
Old question, but just for future reference:
Theorem: Let $k$ be a noetherian ring, $A$ a $k-$coalgebra (coassociative counital) that is flat as $k-$module and $V$ a $A-$comodule. Let $v \in V$ and $(v)$ be the $A-$subcomodule generated by $v$, then $(v)$ is finitely generated as $k-$module.
See Milne's Basic Theory of Affine Group Schemes (page 117, aside 4.7) or Jantzen's Representations of Algebraic Groups (section 2.13) for details. The key idea is using the following lemma:
Lemma: Let $k$ be a ring, $A$ a flat $k-$coalgebra, $V$ a $A-$comodule and $W \subseteq V$ a $k-$submodule. Let $$ W^0 = \Delta_V^{-1}(W \otimes_k A) = \{v \in V \,|\, \Delta_V(v) \in W \otimes_k A\} $$ Then $W^0$ is a $A-$subcomodule and $W^0 \subseteq W$.