I'm studying Artin's Algebra right now, and I'm stuck from the beginning on his proof of the Jordan Decomposition Theorem. It goes somewhat like this:
Theorem: Let $T$ be a linear operator on a finite-dimensional complex vector space $V$. There is a basis B of $V$ such that the matrix of $T$ with respect to B has Jordan form.
Proof. Induction on the dimension of $V$ allows us to assume that the theorem is true for the restriction of $T$ to any proper invariant subspace. So if $V$ is the direct sum of proper $T$-invariant subspaces, say $V_1 \bigoplus \space . . .\space \bigoplus V_r$, with $r > 1$, then the theorem is true for $T$.
Artin then goes on to complete a (pretty long) proof of the theorem, but I can't even understand this first step; how does induction on the dimension of $V$ even allow us to assume anything? Can someone please clarify this for me? Thank you in advance!