1 2 is not a positive integer even though both 1 and 2 are positive integers. For example the positive integers are closed under addition but not under subtraction. In mathematics a set is closed under an operation if performing that operation on members of the set always produces a member of that set.
Your write-up is correct insofar as it goes but it helps to be clear WHICH closure axiom you are invoking because a vector space has several associated with it.
By closed under addition we mean that if r. As in must the first element of the second line be negative or it can be prositive also. First Ill show that nZis closed under addition. We prove commutativity a b b a by applying induction on the natural number bFirst we prove the base cases b 0 and b S0 1 ie.