Consider the following theorem. The product of any even integer and any odd integer is even. Here is a proof of the theorem with at least one incorrect step. Suppose m is any even integer and n is any odd integer. If m · n is even, then by definition of even there exists an integer r such that m · n = 2r. By definition of even and odd, there exist integers p and q such that m = 2p and n = 2q + 1. Therefore, by substitution, the product m · n = (2p)(2q + 1) = 2r. But r is an integer by the statement in Step 2. Thus m · n is two times an integer, so by definition of even, the product is even. Identify the mistakes in the proof.

Step 1 begins an argument by example using specific values m and n
Step 2 assumes what needs to be proved about m n at the end.
Step 3 should use only one variable for an integer to write m and n as even and odd.
Step 4 asserts an equivalence of expressions that is not actually known to be true.
Step 5 relies on a previous incorrect step that assumes what needs to be proved
Step 6 draws a conclusion about m n that belongs in the hypothesis