Of course, the Cayley Hamilton Theorem is not really hard, and there are many many proofs of it. But you have to admit that, at least when you first step into linear algebra, it's rather surprising that it's enough to proof the theorem for diagonal matrices (which is a very short calculation). Because then you can derive it for diagonalizable matrices, which are dense with respect to the Zariski Topology (assuming w.l.o.g. that the ground field is algebraically closed). The latter is because every non-empty open subset is dense, a rather strange but here very useful property.

The same procedure applies to other polynomial identites in linear algebra, for example that the characteristic polynomials of $AB$ and $BA$ coincide.