Here's a sketch of a proof.
Let U and V be the subspaces. If U≠V then neither of U and V can contain the other. So there exist vectors and . Then is in neither U nor V. If U=V then take to be any vector not in U.
Now proceed by an inductive construction. Let be the subspace spanned by U and , and let be the subspace spanned by V and . These spaces have the same dimension as each other. If they are not the whole space then we can repeat the construction in the previous paragraph to get a vector that is in neither nor . Continue doing this, getting a sequence of vectors . On the assumption that the ambient spaces is finite-dimensional, this construction will stop after a finite number of steps. The subspace spanned by will then be complementary for both U and V.