Thread: prove closure under multiplication for integers

1. prove closure under multiplication for integers

i am just starting to learn proofs on my own. this is probably a basic proof but i am having trouble with it. so far i am thinking of using induction to prove this. the problem is if a and b are in the set of integers, prove that ab is also in that set. so i'm letting a be a fixed integer and S be the set of all integers b such that ab is in the set of integers. its obvious that 1 is in the set S. i'm trying to show that b and b+1 are in the set as well. is this the correct method to prove the statement?

2. before you go there - you need to be sure of a few things (so that you know what are you proving and what are you using to generate your proofs)
1. What is the definition of "set of integers"? What are the properties/axioms of this set (for e.g. induciton)?
2. How do you define '+'? What are the properties/axioms on '+'?
3. How do you define '.'? What are the properties/axioms on '.'?

If you have not thought about these, I would suggest you spend a little time in getting a grasp of the above first and then attempt your problem