Let $T$ be a linear operator on a finite-dimensional complex vector space $V$.
(a) Let $\alpha$ be an eigenvalue of $T$, and let $V_{\lambda}$ be the set of generalized eigenvectors, together with the zero vector. Prove that $V_{\lambda}$ is a $T$-invariant subspace of $V$.
(b) Prove that $V$ is the direct sum of its generalized eigenspaces.
This is a problem from Artin, chp 4. I've understood how to prove part a), but I can't figure out how to show it for all generalised eigenspaces.