Let be non-empty subspaces of V. Prove that if is a subspace of V, than either (I don't know how to do this in Latex) or . ( symbol should mean proper subset here).
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This should be done by contradiction I think. So assume that there exists and , wuch that .
Still assume now that is a subspace of V.
Now I'm having a brain fade. Hint perhaps?
Suppose, and then there's a vector, call it
There's also
We have so if the union is a subspace we should get
However and since would imply ( by definition of subspace, summing ) that analog. with
So absurd
So our assumption and , must have been wrong
If one is contained in the other, the union is obviously a subspace, and so we are done