The Cayley-Hamilton theorem. Apparently, Cayley only proved it for $2\times2$ and - in a horrendous calculation - for $3\times3$ matrices and then wrote something outrageous in the spirit of "and similarly, we can prove it for any $n$". Hamilton then proved another special case in a paper on linear operators on the space of quaternions. Nowadays, it is proven in full generality in just a couple of lines, using the fact that the set of diagonalisable matrices of a given dimension, for which the theorem is trivially true, is dense in the set of all matrices of the same dimension.
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