Irreducible implies finite-dimensional for finite group

Statement
Any irreducible linear representation of a finite group (over any field) is finite-dimensional. In fact, its degree (i.e., dimension of the vector space) is at most equal to the order of the group.

Related facts
In fact, the dimension of a vector space admitting an irreducible representation of the group (also called the degree of the irreducible representation) is subject to many constraints, particularly when the field is a splitting field for the group, for instance, when the field is $$\mathbb{C}$$.

Proof idea

 * Pick a nonzero vector
 * Consider the vector subspace spanned by the orbit of this vector under the action of the group
 * Prove that this vector subspace is nonzero, finite-dimensional and is invariant under the group action
 * If the original representation was irreducible, this subspace must equal the whole space, so the whole space is finite-dimensional.