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Okay, cool. Your english is very good for a non-native speaker.
If you've been reading this forum for a while, you might have learned that I don't generally just prove other people's theorems for them, especially if it's an assignment! I honestly want you to learn this stuff on your own, right? On the other hand, if you're simply stuck, staying stuck won't teach you anything so here's a compromise. I'll prove to you a really useful lemma that you can use to prove your theorem, okay? Here it is...
lemma: If U and W are vector spaces and the union of U and W is also a vector space, then one of them must be a subspace of the other.
proof: Suppose that neither U nor W are subspaces of each other. Then, by definition, there must exist vectors u from U and w from W such that w is not a member of U and u is not a member of W.
Because the union of U and W is a vector space, the sum u + w must also be in this union. By definition, this means that u + w is either in U or in W. However, if it's in U then u + w - u must also be in U, but u + w - u = w, which is not supposed to be in U! This is a contradiction, so u + w must be in W.
Folowing the same logic the other way, you can deduce that u + w is not in W either, so the whole situation is a contradiction. If no such u and w can exist, then one vector space must necessarily be a subset, and therefore a subspace, of the other.
QED.
I hope this helps! If it doesn't, come back and I'll show you how this lemma applies to your problem...
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