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**dwsmith** Let F be a an infinite field. Prove that a vector space, V, over F cannot be a finite union of proper subspaces.

I have already shown that the union of two proper subspaces is a subspace iff. one is contained in the other. Let's call this lemma 1.

Can I use lemma 1 and prove this by induction?

P(2): proves the union may not even be a subspace.

Assume P(k) is true for a fixed by arbitrary integer k that is greater than or equal to n, i.e. P(k) isn't a subspace.

Then, by lemma 1 and the inductive hypothesis, P(k+1) is just the union of P(k) and the k+1 subspace which may not even be a subspace. Therefore, since the union may not be a subspace, the finite union of proper subspaces cannot be a vector space over F.

Does this work?