1. The problem statement, all variables and given/known data n * m = m * n where m, n are natural numbers. 2. Relevant equations I am working from Terrence Tao's class notes and he includes 0 in the natural numbers. m++ stands for m+1. He calls it incrementation and uses it to explain the rules of addition. 3. The attempt at a solution Proof. We will induct on m holding n constant. For the base case, let m = 0. Then we have n * 0 = 0 * n. Both sides of the equation equal to 0. Now we assume inductively that n * m = m * n. For the inductive step we need to show that n * (m++) = (m++) * n. Since m is a natural number, we know from a previous proposition that m++ is also a natural number. Thus we can choose q to be the successor of m or in other words q = m++. Then we have n * q = q * n. According to the inductive hypothesis, this is true. This closes our induction. Can someone take a quick look at my proof and tell me if there is anything wrong with it? Thanks!