Having established the necessary foundation, we’re ready for proof techniques. We examine several techniques, including direct proof, proof by contradiction, proof in cases and working backwards. We emphasize Pólya’s list throughout by understanding the problem we are given, devising a plan to solve it, carrying out our plan, and then looking back at the solution to see that it is correct and to see what we have learned from the solution. We illustrate the method using two of G. H. Hardy’s historical examples. In Tips on Definitions we explain the importance of understanding definitions before you memorize them. The tips emphasize our belief that providing examples and nonexamples enhance one’s understanding of the ideas behind the new definitions.