Triangulability theorem

Statement
Let $$k$$ be an algebraically closed field. Denote by $$GL_n(k)$$ the general linear group of invertible $$n \times n$$ matrices over $$k$$ and by $$B_n(k)$$ the Borel subgroup: the subgroup of invertible upper triangular matrices. Then, $$B(n,k)$$ is conjugate-dense in $$GL(n,k)$$. In other words, given any matrix in $$GL(n,k)$$, we can conjugate it (or change basis) to make it upper triangular.

This is an immediate corollary of the Jordan canonical form theorem, though it can be proved independently as well.

Note that the result fails over non-algebraically closed fields; in particular, it fails when $$k$$ is a finite field.