18 Introduction

Real Analysis is a course that brings together the foundations from calculus and abstract mathematics. In calculus we gained a deep understanding of sequences and series as well as integration and the use of limits within all three of these topics. We gained an in depth understanding of convergence and divergence as well in calculus and when we moved on to abstract mathematics we became well versed in sets and set theory including functions of sets, properties of sets and relations.

In Real Analysis we took everything we previously learned and went even further. This course truly is analysis, and you walk away with such an in depth understanding of how mathematics works and why. Almost every aspect of the course involves proving concepts and theorems that are part of understanding the inner workings of real numbers, sets and sequences, convergence and limits to name a few.

In previous courses we are given rules and theorems meant to help us be able to use math concepts in our calculations. These courses all required us to show our mastery of utilizing integrals, derivatives, limits, etc. In real analysis, we analyze the inner workings of the previously mentioned concepts in order to understand why they work the way they do and why theorems and laws exist for us to trust when we are attempting to run a calculation.

Without understanding why things work we will never be able to truly master anything. With this understanding in mind we will briefly cover each of the topics covered in a real analysis course.