 It is quite common in mathematics to use inductive reasoning for proofs of the sort where you are trying to prove something is true for all numbers in an infinite set. Proving the commutivity property for addition a plus b equals b plus a for all counting numbers is one of them. All we have to go on are our postulates and one proven theorem. The postulates are, one is a number, every number n has one and only one successor number, n plus one, no two different numbers have the same successor number, and the associative property of addition a plus b plus c is equal to a plus b plus c. The commutative property proof has two parts. In the first part we show that a plus b equals b plus a for any value of a when b equals one. Our inductive assumption is that for some value of a the relationship is true. We will show that that implies that the relationship is then true for a plus one. We want to show that a plus one plus one is equal to one plus a plus one. We start with a plus one plus one. We can set it equal to one plus a plus one by our inductive assumption and we can change that to one plus a plus one by the associative property and we are done. Now we will show that a plus one equals one plus a when a equals one. We start with a plus one, substitute in one for a, and then substitute in a for the other number one and we are done. So it is true for a equals one and by the first result we know that it is true for a equals one plus one, that's two, and then for two plus one, that's three, etc. for all counting numbers. So in part two our inductive assumption is that a plus b equals b plus a for some value of b. We will show that this implies that a plus b plus one is equal to b plus one plus a. So we will start with a plus b plus one. We can use the associative property around a and b. We can then use the inductive assumption to get b plus a plus one, reuse the associative property to get b plus a plus one, and then use the results of part one that showed a plus one was equal to one plus a to get b plus one plus a and then use the associative property again to get b plus one plus a and we are done.