Schedule: (usually) Tuesdays and Thursdays, 11:00A-12:20PM or 1:30PM-2:50PM. 216 is offered in the Fall semester only, and it continues with 218 in the Spring. Both courses include optional evening review sessions run by undergraduate course assistants on a schedule that is set up after classes begin.

Brief Course Description: Rigorous theoretical introduction to the foundations of analysis in one and several variables: covers basic set theory, vector spaces, metric and topological spaces, continuous and differentiable mappings between n-dimensional real vector spaces. The textbook, An Introduction to Analysis by Robert C. Gunning, was developed in conjunction with the course.

Who takes this course? This course is suitable both for prospective mathematics majors and also for non-majors who arrive at Princeton with substantial prior experience and interest in proof-based mathematics at the university level. Often the students in 216-218 are considering various majors including mathematics, physics, engineering with a strong theoretical component, computer science, or philosophy. Many prospective math majors have deep interests in the humanities as well, and at least fleetingly contemplate majoring in history or music or classics, for example. This course is meant to show by direct experience how mathematicians think and build a theory and determine what is true. It also satisfies the multivariable calculus and linear algebra prerequisites for math, physics and engineering, albeit in a very theoretical and abstract form.

Prerequisites and Placement Information:  A very strong aptitude for mathematics and genuine mathematical curiosity are essential. Do you love mathematics? In high school did you already learn how to construct a mathematical argument by dividing a question into logical steps where each step is explained and justified carefully, giving references if necessary? Have you already been reading math books that are organized in terms of abstract definitions and formal theorems and rigorous proofs, where the reader is expected to construct examples and counterexamples independently and the problems are about proving further abstract statements using the theorems and definitions from the chapter?

Students’ prior experience with proofs may be via university level analysis courses taken in high school, through independent reading and study or via extracurricular activities like math camps or Olympiad training and competition. Prospective math majors without this extensive background should plan to take MAT215-217 instead.

If you are not sure whether to take 215 or 216, you can begin by e-mailing the departmental undergraduate administrator (Michelle Matel, indicating your interest in 216. In addition to describing your mathematics background, you should also tell us about you specific experience with proofs in the past, and what sort of proof-based homework problems you have been assigned. She will forward the information to the 216 instructors. If 215 is the right course for you, you will learn more and build a more solid foundation for your upper division work in mathematics or related disciplines. You can also speak with Dr. McConnell, our Junior Advisor.



  • How hard should I expect to work in this class?  This class is pretty intense, but the work load can vary quite a bit for individual students, depending on your background and your goals. We expect that you will need to spend substantial time outside of class, reading the textbook and reviewing your notes in order to understand the proofs and the definitions from the textbook well enough to be able to adapt them to create new proofs and construct counterexamples. Following up in office hours or at the undergraduate-led problem sessions or with your study group will likely be important for understanding all the new ideas. The problems require both creativity and knowledge, and so you will need to spend unpredictable amounts of time in active contemplation while you wait for inspiration or the right insight. All in all, you should be ready to spend at least 10 hours per week working outside of class, and we hope you will love every second of it! 
  • You are really not sure about a major, but you love math and you are good at it. You have never taken a proof-based class before. You should not take 216 unless you have a solid background with proofs. You will learn a lot in 215 or possibly 203.
  • You have never had a course with rigorous proofs but you know you want to be a math major. Please take 215 instead. You will learn much more. And keep an open mind. It is normal and good to get excited about a major that you never even considered until you took a university course that showed it to you in a new light. Maybe 215 will show you that you love math even more than you realized. But it is also possible that 215 will show you that proof-based math does not appeal to you right now. Follow your interests, and be willing to change your plans as you get more information. We want you to be happy.
  • The placement information suggests that 215 is the right course for you, but it is an ‘extra course’. It doesn’t satisfy the prerequisites for physics and engineering and you might end up majoring in one of those instead. So you would rather take 216-218 instead of 215.  Trying to do 216 without the expected background experience with proofs will be a very stressful experience and can leave you feeling burnt-out and exhausted, and frustrated that you don’t have enough time to really understand the ideas with the depth they deserve. Moreover, it may make it very difficult for you to keep up in your ambitiously chosen physics and computer science courses, or force you to give up extra-curricular activities that give your life meaning and joy. It is not worth it! If 215 is the right course, take 215. And if you are concerned about satisfying prerequisites, you can always sit in on both 215 and 203 for a few classes until you decide which is a better choice for you.
  • You talked to some math majors and they told you that all the serious math majors take 216. Sigh. Many strong majors and future mathematicians start in 215, and sometimes they even start in 103. Math majors can give good advice, but it is based on their own experience and it may not be applicable to you. Other advisers have a broader perspective and their advice is also worth considering. Don’t confuse fast thinking with deep thinking. Real math majors make up their own minds and hardly ever listen to advice, and sometimes it works out just as planned and sometimes they have regrets. Enthusiasm is good, but so is realism and flexibility. Think about all the information available to you as you make your preliminary decisions, and be ready to adjust as you learn more in the first few weeks of the semester if necessary.  
  • You have quite a bit of experience with proofs from independent reading or university-level analysis courses you took in high school or from extracurricular activities like math camp. Many of the topics for the course are already familiar to you. Maybe you should start at the 300-level instead?  Try the Sample Problems. If you know how to do these problems already, then skipping 216 might be reasonable, although it is rare. Even if you have seen these ideas before, most people can still learn quite a bit from contemplating them further. Incoming students often over-estimate their mathematical knowledge and it is difficult to know what you don’t know. If after trying the sample problems you still feel the same way, you will need to talk to the Director of Undergraduate Studies or to the 216 instructors and convince them as well. First-year students can sign up for 300-level courses only with permission from the department.
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