# Real Analysis

Here, you find my whole video series about Real Analysis in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

### Introduction

Real analysis is a video series I started for everyone who is interested in calculus with the real numbers. It is needed for a lot of other topics in mathematics and the foundation of every new career in mathematics or in fields that need mathematics as a tool:

With this you now know the topics that we will discuss in this series. Some important bullet points are limits, continuity, derivatives and integrals. In order to describe these things, we need a good understanding of the real numbers. They form the foundation of a real analysis course. Now, in the next video let us discuss sequences.

### Let’s get started

The notion of a sequence is fundamental in a lot of mathematical topics. In a real analysis course, we need sequences of real numbers, which you can visualise as an infinite list of numbers:

Now you know what a convergent sequence is. However, not all sequences are convergent. A weaker property is the notion of a bounded sequence.

At this point you know a lot about sequences, especially about convergent sequences. Since we do not want to work every time with the definition, using epsilons and so on, we prove the following limit theorems:

Another important property we will use a lot for showing that a sequence is convergent and also for calculating its limit is the sandwich theorem:

Now, we go back to general subsets of the real numbers and talk about some important concepts, supremum and infimum of sets:

Let us talk about Cauchy sequences. These special sequences and the concept of completeness are deeply connected.

Very good! It is time to explicitly calculate with an example. Also it is a good time to introduce, the very famous, Euler’s number.

Another important topic in Real Analysis and for sequences are so-called accumulation values.

By knowing what accumulation values for sequences actually are, we can discuss a famous and important fact in this field: the Bolzano-Weierstrass theorem.

There are two special accumulation values for a sequence: the limit superior and limit inferior.

Let us do some examples and calculations rules for the limit superior and limit inferior:

Now, we are ready to talk about some important notions for subsets of the real numbers. Namely, we discuss what open, closed, and compact sets actually are.

Since we now know what compact sets in the real numbers are, we can ask what are necessary and sufficient conditions for knowing that a given set is compact. Indeed, for subsets of the real number line, the famous Heine-Borel theorem gives us a nice description:

Let us start with the next big topic: series. One can see them as special sequences but we will see that they occur often in different problems.

Two important examples for series are discussed in the next video: geometric series and harmonic series:

In the next videos we will talk about a lot of criteria we can use to test for convergence of a given series. We start with the simplest one: the Cauchy criterion:

The next criterion we will talk about is very useful for alternating series and called the Leibniz criterion:

Since we already know some convergent and divergent series, it might be useful to use them to decide if a given series is also convergent or divergent. This is known as the comparison test. One distinguish between the majorant criterion and the minorant criterion, depending from which side one looks at the series and if one wants to show convergence or divergence.

By using the geometric series, the majorant criterion immediately leads to two very helpful tests: root test and ratio test.

A natural question when dealing with a series is if one can reorder it without changing the value like one knows happens for ordinary sums. However, for series a reordering can change the limit:

Let’s close the chapter about series with an important operation: the Cauchy product:

The next chapter will deal with continuous functions. However first, we need to define some notions for sequences of functions:

The notion of pointwise convergence for a sequence of functions is a very natural one:

Also the notion of uniform convergence for a sequence of functions is important and can be expressed with the help of the supremum norm:

Now let us go to the definition of continuity. For this, we will need the notion of limits for functions:

The definition of continuity can be easily formulated with sequences or just which the limit notion from above. After having this, we can look at some examples:

Now we can finally talk about the Epsilon-Delta definition of continuity.

Continuous functions have a lot of nice properties. We start with the question what happens when we combine different functions, like adding, multiplication, and the composition.

Next nice property: continuous images of compact sets are always compact.

Uniform limits of continuous functions have to be continuous as well:

The intermediate value theorem, finally!

Now, at this point, we really should look at important examples. The exponential function, we already have discussed a little bit. However, it turns out that there is an inverse, we call the logarithm function. Moreover, we can generalise the whole concept and find so-called power series:

After we have looked at some examples, we are now ready to talk about differentiability. This means we will finally define the derivative:

We can prove the sum and product rule for differentiable functions:

Also very important is the so-called chain rule when we deal with compositions of differentiable functions:

In the following videos, we will look at a lot of examples, in particular ones that are given by power series. However, for them, differentiability is not clear at all. To show this property, we have to talk about what the uniform convergence of functions actually conserves. The next video is about such a nice theorem.

Let’s apply all our knowledge about derivatives to get important examples:

Now we want to calculate the derivative of the logarithm function. Since it is the inverse of the exponential function, we should try to find an inversion formula for derivatives. This is indeed possible:

In this an in the next video we will talk about the famous Mean Value Theorem. As a groundwork, we have to proof Rolle’s theorem first:

Now, we are ready to formulate and prove the Mean Value Theorem first:

Next, let us apply our knowledge to prove the popular Theorem of l’Hospital and apply to examples:

Now, let’s formulate a generalisation for l’Hospital’s theorem by considering four different cases that can occur in applications:

The next important topic will be about Taylor’s formula. In order to understand this, we need to talk about higher derivatives first:

Now we finally discuss Taylor’s Theorem:

Let us immediately look at an application for Taylor’s Theorem by approximating a value of complicated function:

Now we can prove Taylor’s Theorem by using the generalised mean value theorem from above:

Finally, we start with the Riemann integral by using the definition of a partition and step function:

Now we can define the Riemann integral for step functions:

Let us show important properties of the Riemann integral for step functions:

Now we are finally ready to define the Riemann integral for bounded functions:

Let us look at examples for calculating the Riemann integral using the approximation by step functions:

Now we can talk about some important properties of the Riemann integral for bounded functions:

Finally, we talk about one of the main results in this course: the so-called First Fundamental Theorem of Calculus:

And also we should talk about the Second Fundamental Theorem of Calculus:

Now let us prove the Fundamental Theorem of Calculus by using the Mean Value Theorem of Integration:

A similar rule that can simplify integrals is given by integration by parts. It can help you when you have an integral of a product of two functions.

The next thing in our toolbox for integration can help you when you need to integrate rational functions. What we will need is the so-called partial fraction decomposition. After doing this, finding the antiderivatives is not hard at all.

The next part is about improper Riemann integrals. In particular, we will consider integration on unbounded domains. This is something the ordinary Riemann integral couldn’t cover by definition. However, we can extend this definition when we combine it with a usual limit of real numbers.

You might have already recognised that these improper Riemann integrals are related to infinite series. Indeed, we find a similar comparison test for integrals as well:

Moreover, integral can be used to show convergence of series and vice versa. However, often integrals are easier to calculate because of the fundamental theorem of calculus. Therefore, we find the integral test of convergence for series:

Now, we can also extend the notion of an improper Riemann integral for functions that have holes in the domain of definition. More precisely, we define the Riemann integral for unbounded functions:

Now, finally, we tackle the last video in this series. Let us close the topic of improper Riemann integrals and also talk about a generalisation of them: the so-called Cauchy principal value: