Lemma 3

Let $\varrho$ be a continuous linear representation of G in a separated topological K-vector space E of finite dimension. If E is nonzero, then there exists an irreducible subrepresentation of $\varrho$.

Since E is finite-dimensional and nonzero, there exists a G-invariant subspace F of E which is nonzero and of minimal dimension. This subspace is closed in E and defines a subrepresentation of E; every subrepresentation of F is also a subrepresentation of E, and the representation F is therefore irreducible by minimality.