This course is a detailed study of the real numbers and real-valued functions, leading to the reworking of the material from first and second-semester calculus in a rigorous setting. We will begin by setting down the axioms necessary to define real numbers and will spend some time studying their properties. We will then be ready to tackle sequences and series using the proper definition of a limit of a sequence and prove some important resuls like the Monotone Convergence Theorem and the Bolzano-Weierstrass Theorem. Next we will talk about the basic topology of the real numbers and cover topics like open and closed sets, connectedness, compactness, and the Heine-Borel Theorem. This will allow us to delve into the study of functions on the real numbers and develop the notions of limits and continuity in a rigorous way. With this in hand, we will look at sequences and series of functions, derivatives, and integrals, and prove some amazing results such as the Mean Value Theorem and the Fundamental Theorem of Calculus.

Most of what we do will be for functions with domain and range the real numbers, but we will often extrapolate to functions between general Euclidean spaces R^m and Rⁿ. Further, there will be instances when the generalization of the material to arbitrary metric spaces does not come at a great cost, so we will keep this in mind and introduce these ideas whenever possible.

The more general objective of this course is to continue providing you with a deeper understanding and working knowledge of mathematics, while in the process strengthening your analytical skills, increasing your ability to communicate mathematics symbolically and orally, making you comfortable with reading and understanding mathematics on your own, and continuing to develop your appreciation for abstract mathematics. In particular, a strong emphasis will be placed on proofs and proof techniques.