This paper discusses ideas for a different approach to teaching the foundations of mathematical analysis. The main idea is to avoid the use of what Keith Devlin in 2005 called “formal definitions,” which are definitions that nobody can understand without working with them. For students without mathematical maturity, these definitions can be difficult to understand and use. This paper discusses an approach that uses the history of mathematics to first develop fundamental concepts and only introduces formal definitions after the concepts are understood. The audience for this approach is third-year undergraduate students. Several examples are provided in the paper.