# Real Analysis

Historically, analysis developed from the need for a continuous number system that could be infinitely subdivided, the real number system [24]. Analysis became necessary when working with and describing parts of the world that could be considered continuous. In actuality, these aspects of nature are not infinite, but they can be subdivided so far into such small pieces that the real numbers are very useful in describing them [24].

When calculus was starting out, many ideas were not very well defined, and there were many logical problems. Analysis grew from a need to fix these problems and to fill in the holes and gaps that calculus still possessed [24]. For example, analysis created accurate definitions for ideas in calculus, such as limits and functions, that up to this point in history had not been well-defined [24]. The mathematicians who developed calculus used the idea of limit of a function or sequence with only an intuitive understanding of what these ideas meant. It was not until the work of the German mathematician Karl Weierstrass that a formal definition for limit was actually given [1].

Analysis is a mathematical branch that can be seen in many sciences. But it is also used in economics, finance, sociology, and similar areas [24]. Because analysis is so interconnected with calculus, it has many of the same applications, such as dealing with motion and instantaneous rates of change [24]. Analysis, along with calculus, is used to find the length of a curve or the area under a curve. This type of problem has many applications in the real world. For example, finding the size of an irregularly shaped piece of land, finding the mass of a curved object, or calculating the amount of paint needed to cover an irregularly shaped surface [24]. Interestingly enough, the same methods applied to the situations just mentioned can also be applied to distance traveled when speed is not constant or the amount of fuel consumed by a vehicle [24]. Other applications of analysis and calculus include finding how steep a hill is or how fast a disease-causing organism can spread through the population [24]. Obviously, these are not the only applications, but they provide a glimpse into the kind of real world problems that analysis, working with calculus, can solve. These applications also give an idea of how interconnected calculus and analysis are. One cannot work at an optimal level without the other.

We are going to start our discussion with some axioms of the real numbers. We will begin by introducing the field and ordered field axioms, all of which are satisfied by the real numbers. We will then take a brief look at the triangle inequality, which is a simple but very useful statement. After the axioms, we will move on to the topology of the real numbers. We will begin with the Axiom of Completeness and see how this leads us to the density of the rational and irrational numbers in the reals. Next, we will introduce open and closed sets, along with compact sets. We will then see how closed, bounded sets are connected to compact sets through the Heine-Borel Theorem.

After these topics, we will go back to some ideas we discussed in calculus, sequences, limits, and continuity. We will discuss Cauchy sequences and subsequences and see how they help with determining convergence of a sequence. We will introduce the epsilon-delta definition of limits and see how this definition, along with calculus, can help us find limits. We will then take a look at the Intermediate Value Theorem and wrap up this section with continuity. We will look at an alternate form for the definition of continuity, which can be derived from the epsilon-delta definition and the calculus definition of continuity. Finally, we will look at uniform continuity and see how to prove whether a function is uniformly continuous or not. So, without further ado, let’s begin our discussion of the interesting ideas and concepts of mathematical analysis.