Here are a few general comments, inspired by some observations on the homework:

• In order to prove that a set A is contained in a set B, prove that every element of A is actually contained in B.
• In order to prove equality between two sets A and B, you typically need to show that A&subB and that B&subA.
• Part of proving that a subset S&subV is actually a subspace is showing that S is nonempty. Typically, the easiest way to do this is to show directly that 0&isin S.