Real Analysis I (saylor.org)

In calculus, you learned to find limits, and you used these limits to give a rigorous justification for ideas of rate of change and areas under curves. Many of the results that you learned or derived were intuitive – in many cases you could draw a picture of the situation and immediately “see” whether or not the result was true. This intuition, however, can sometimes be misleading.
In the first place, your ability to find limits of real-valued functions on the real line was based on certain properties of the underlying field on which undergraduate calculus is founded: the real numbers. Things may have become slightly more complicated when you began to work in other spaces. For instance, you may remember from multivariable calculus (calculus in three or more real variables) that for some functions there were points where some directional derivatives existed and others did not. In fact, there exist other more exotic spaces where other complications arise.
In the second place, the techniques that you used to find limits may have been very informal. In this course, you will learn to rigorously justify every step in the limiting process or proof. Learning to do this well in the familiar context of the real line, will prepare you for wilder, more complicated mathematical situations. After a brief review of set theory, you will dive into the analysis of sequences, upon which all analysis of Euclidean space (and any separable metric space) is based.
Course Requirements:
Have completed Single-Variable Calculus I, Single-Variable Calculus II, Multivariable Calculus, Linear Algebra, and Differential Equations from “The Core Program” in the math discipline, or their equivalents. Introduction to Mathematical Reasoning is recommended for students lacking experience with rigorous methods of proof.