Prove the sum of any two rational numbers is rational

Take two integers, r and s.

We can write r as a/b for integers a and b since a rational number can be written as a quotient of integers
We can write s as c/d for integers c and d since a rational number can be written as a quotient of integers

Add r and s:
r + s = a/b + c/d

With a common denominator bd, we have:
r + s = (ad + bc)/bd

Because a, b, c, and d are integers, ad + bc is an integer since rational numbers are closed under addition and multiplication.

Since b and d are non-zero integers, bd is a non-zero integer.

Since we have the quotient of 2 integers, r + s is a rational number.