I'm trying to prove x is a real number is closed under addition and 3Z is closed under addition, where Z is the set of integers.
To prove 3Z is closed under addition, we need to show 3*k_1 + 3*k_2 belongs to 3Z for arbitrary k_1 and k_2 in Z.
Since an interger ring Z distributes multiplication over addition, 3*k_1 +3*k_2 = 3*(k_1 +k_2).
Since k_1 and k_2 belongs to Z which is closed under addition, we can pick k_3 in Z satisfying k_3 = k_1 + k_2.
Now, 3*k_1 +3*k_2 = 3*(k_1+k_2) = 3*k_3, where k_1, k_2, k_3 belongs to Z.
Thus, 3Z is closed under addition.