This problem is from Sheldon Axler's Linear Algebra Done Right, Chapter 8. Let $T \in \mathcal{L}(V)$, where $V$ is a finite dimensional complex vector space. If the matrix of $T$ is upper triangular with respect to any basis of $V$, the number of times $\lambda$ appears on the diagonal of this matrix equals the (algebraic) multiplicity of $\lambda$ as an eigenvalue of $T$.
It suffices to show that $\dim \text{null } T^n = \dim G(0, T)$ equals the number of $0$'s on the diagonal, where $G(0, T)$ is the space of generalized eigenvectors of $T$ with respect to $0$.
I found this rather nice proof on this blog, which uses induction on the dimension of $V$.
I'm interested in finding alternative ways to prove this statement, which may provide a different way of looking at it.