You should read the Handouts as the are mentioned in the lectures. The supplimental reading you can do as you wish.

Handouts

• Algebraic Systems
• Properties of the Real Numbers
• Unit Circles in ℝ²
• Bernstein Polynomial Approximations

Supplemental Reading

• Berkeley's Attack on the Infinitesimal, by Mark Monti (used with permission).
• The Mathematical Shape of Modernity, by Paula E. Findlen, Chronicle of Higher Education, June 23, 2014. (Can only be viewed on a University computer or with a subscription.)
• Derivation of the Trapezoidal Rule Error Estimate
• Cantor's Other Proofs that R Is Uncountable, by John Franks, Mathematics Magazine, Vol. 83, No. 4 (October 2010), pp. 283-289.
• A pedagogical history of compactness, by Manya Raman-Sundstrom, preprint.
• The emergence of open sets, closed sets, and limit points in analysis and topology, by Gregory H. Moore, Historia Mathematica, Volume 35, Issue 3, August 2008, Pages 220--241.
• Stirling's Formula, by Keith Conrad.
• Zeno's Paradoxes, by Bradley Dowden, in the Internet Encyclopedia of Philosophy, a Peer-Reviewed Academic Resource.
• The Continuum Hypothesis, by Peter Koellner, Stanford Encyclopedia of Philosophy, May 22, 2013.

Other textbooks on Real Analysis

• Principles of Mathematical Analysis, by Walter Rudin. (400 level)
• Real and Complex Analysis, by Walter Rudin. (500 level)
• Understanding Analysis, by Stephen Abbott. (300 level)
• The Elements of Real Analysis, by Robert G. Bartle. (400 level)
• Real Analysis, by Halsey Royden. (500 level)
• Foundations of Analysis, by Edmund Landau. (Any level)

Books every serious student of mathematics should have

• The History of the Calculus and Its Conceptual Development, by Carl B. Boyer Amazon Dover Press B&N
• Proofs and Refutations: The Logic of Mathematical Discovery, by Imre Lakatos. See especially Appendices I and II. MAA Amazon