I think you mean "the set of all b such that Ax= b for some x in " - that is known as the image of and is also the "column space" since it is spanned by the columns of the matrix.
To prove any subset is a subspace, it is sufficient to show that it is closed under addition and scalar multiplication- or, combining the two, that if u and v are in the subset and a and b are scalars, then au+ bv is also in the subset.
If u and v are in the subset then there exist x and y in such that Ax= u and Ay= v.
Look at A(au+ bv).