# Real Analysis l

An introduction to the theoretical basis for the calculus, with an emphasis on rigorous proof. Properties of the real number system; sequences and series; continuity; elementary topology of the real line, Euclidean space and metric spaces; compactness; pointwise and uniform convergence.

Prerequisite: Mathematics 204, 220 and 251.

1 unit — Agbanusi, Moran

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### Real Analysis

Real Analysis aims to rigorously prove much of what is seen in Calculus. Students will develop a deeper understanding of several concepts that serve as the foundation for Calculus.
One of the main goals of Real Analysis is to rigorously prove much of what we see in Calculus. We must first have a basic understanding of the real line, as this is the underlying space that we are studying. The set of real numbers consists of rational numbers (those that can be expressed as a fraction) and those that cannot, called irrational numbers. However, there are many interesting and important questions to ask and facts to prove about the real line. For example, are there more irrational numbers than rational numbers? Does the interval (0,1) have the same size as the entire real line? Can you always find a rational number between two irrational numbers? The course begins by studying properties of the real line and giving answers to these questions. Armed with a deeper knowledge of real numbers, we can then study limits, as these are the foundation for all of calculus. We examine limits of sequences of real numbers, limits of real-valued functions and the relationship to continuity, and why we would want to study limits of sequences of real-valued functions. Along the way, we will see interesting examples that challenge some of our preconceptions and prepare you for the study of differential and integral calculus in Real Analysis 2. The course concludes with an introduction to the topology of metric spaces so that we can generalize many of the concepts learned throughout the block to study spaces other than the real line.

## Offerings

Term Block Title Instructor Location Student Limit/Available Updated
Fall 2021 Block 1 Real Analysis l Molly Moran Tutt Science Building 221 12 / -1 09/17/2021
Spring 2022 Block 6 Real Analysis l Ike Agbanusi TBA 25 / 25 09/17/2021
Report an issue - Last updated: 09/17/2021