Course Description

This course provides a rigorous foundation for understanding the real number system and the fundamental concepts underlying calculus. We will carefully examine the ideas you may have encountered intuitively in calculus, limits, continuity, sequences, and series, and develop them with mathematical precision and proof.

Analysis is where mathematics transitions from computation to careful reasoning about why mathematical statements are true. You'll learn to think deeply about questions like: What does it really mean for a sequence to converge? How do we know that certain numbers exist? What makes a function continuous, and why does continuity matter?

Through this course, you'll develop the ability to read, write, and construct mathematical proofs while exploring the elegant structure of the real numbers. This is not just about learning theorems, it's about understanding the logical architecture that makes calculus work and acquiring the mathematical maturity needed for advanced mathematics.

Topics include: the construction and properties of real numbers, sequences and their limits, series and convergence tests, limits and continuity of functions, and the topology of the real line. By the end of this course, you'll have a deep appreciation for the precision and beauty of mathematical analysis.