Not a very big theorem, but the rather cute fact that the characteristic polynomials of $AB$ and $BA$ coincide ($A,B$ some $n \times n$ matrices, say complex).
It's true if $A$ is invertible since then $AB = A(BA)A^{-1}$ and similar matrices have the same characteristic polynomial. By the density of $GL(n,\mathbb{C})$ in $Mat_n (\mathbb{C})$, the result follows in general.