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?