https://maximummathematics.com/books/foundational-mathematics/proof/mistakes-in-proofs/index.htmlSo far, we’ve seen three different proof techniques: one direct, and two indirect. Applying any of them requires close adherence to the rules of inference discussed throughout this chapter. However, if we make an argument that uses an invalid inference rule, we have an invalid argument, and hence an invalid proof. In this section, we discuss a few of the most common types of errors that can be made. Violating Hypotheses of a Theorem or Axiom Remember that a theorem guarantees some result holds when a certain collection of premises are satisfied. If even one premise fails to hold in a given scenario, the theorem no longer applies — its conclusion may still happen to be true, but not because of the theorem itself. Consider the following “proof” that $1 = 2$:Hugoen-US