Proof.
Suppose the numbers \(a\) and \(b\) are even. This means that \(a = 2k\) and \(b=2j\) for some integers \(k\) and \(j\text{.}\) The sum is then \(a+b = 2k+2j = 2(k+j)\text{.}\) Since \(k+j\) is an integer, this means that \(a+b\) is even.
Suppose the numbers \(a\) and \(b\) are even. This means that \(a = 2k\) and \(b=2j\) for some integers \(k\) and \(j\text{.}\) The sum is then \(a+b = 2k+2j = 2(k+j)\text{.}\) Since \(k+j\) is an integer, this means that \(a+b\) is even.
Notice that since we get to assume the hypothesis of the implication, we immediately have a place to start. The proof proceeds essentially by repeatedly asking and answering, “what does that mean?” Eventually, we conclude that it means the conclusion.