I'm guessing you're allowed to assume that the product of two integers is still an integer. Then you can indeed say that is a rational number, i.e. the ratio of two integers.
Here I go again with these proofs, this time I am trying to prove the sum of two rational numbers is rational.
This is what I have...
Assume x is rational
Assume y is rational
Let x=a/b where a and b are integers and b does not equal 0
Let y=c/d where c and d are integers and d does not equal 0
x+y=a/b+c/d=(ad+bc)/bd
Let w=ad+bc
Let v=bd
We check that w and v are integers which they are and the integers are closed under addition and multiplication
Therefore the sum of two rational numbers is rational