*Here, you find my whole video series about Real Analysis in the correct order and you can choose between the bright and dark version of the videos. I also help you with some text and explanations around the videos. If you want to test your knowledge, please use the corresponding quiz and exercise sheet after watching the video. Moreover, you can also consult the PDF version of the video if needed. In the case you have any questions about the topic, you can contact me or use the community discussion in Mattermost and ask anything. You can also find a quick summary of the course. However, without further ado let’s start:*

#### Part 1 - 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:

###### Content of the video:

00:00 Introduction

00:27 Topic of real analysis

01:31 Requirements

02:05 Axioms of the real numbers

03:54 Credits

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**.

#### Part 2 - Sequences and Limits

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:

###### Content of the video:

00:00 Introduction

00:15 Definition of a sequence

01:50 Examples

06:06 Definition of convergence and divergence

08:55 Example and Archimedean property

11:56 Credits

#### Part 3 - Bounded Sequences and Unique Limits

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

###### Content of the video:

00:00 Intro

00:15 An example for showing that a sequence is divergent

03:41 Definition of a bounded sequence

04:36 A convergent sequence is also bounded

06:09 The limit of a convergent sequence is uniquely given

08:59 Outro

#### Part 4 - Theorem on Limits

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**:

###### Content of the video:

00:00 Intro

00:18 Limit of a sequence

01:16 Theorem on limits

03:48 Example

06:22 Outro

#### Part 5 - Sandwich Theorem

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**:

###### Content of the video:

00:00 Intro

00:10 Limit theorems

00:50 Monotonicity of the limit

01:45 Sandwich Theorem

02:33 Proof of the Sandwich Theorem

05:30 Example

07:48 Outro

#### Part 6 - Supremum and Infimum

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

###### Content of the video:

00:00 Intro

00:14 Intervals

01:31 Maximum and upper bounds

02:19 Minimum and lower bounds

02:44 Definition of bounds

04:14 Examples

05:43 Definition Supremum and Infimum

08:40 Credits

#### Part 7 - Cauchy Sequences and Completeness

Let us talk about **Cauchy sequences**. These special sequences and the concept of **completeness** are deeply connected. So here we need to recall the completeness axiom of the real numbers. Moreover, we also introduce the concept of **Dedekind completeness**, which uses the supremum of subsets.

###### Content of the video:

00:00 Intro

00:14 Convergent sequences

00:54 Different property of a sequence

02:09 Definition Cauchy sequence

02:16 Connection to convergent sequences

03:14 Dedekind completeness

03:47 Sketch of proof

07:35 Application for monotonic sequences

08:44 Credits

#### Part 8 - Example Calculation

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**.

###### Content of the video:

00:00 Intro

00:21 Recalling the monotone convergence criterion

00:55 Introducing the example

01:21 Proving monotonicity

04:49 Proving bounded from above

08:55 End result of the example

09:46 Credits

#### Part 9 - Subsequences and Accumulation Values

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

###### Content of the video:

00:00 Intro

00:13 Definition of subsequences

01:40 Example

03:26 General fact about subsequences

04:00 Example with a divergent sequence

05:04 Definition accumulation value

06:49 Alternative definition of accumulation values

07:47 Credits

#### Part 10 - Bolzano-Weierstrass Theorem

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

###### Content of the video:

00:00 Intro

00:20 Bolzano-Weierstrass theorem

01:13 Proof

05:42 Credits

#### Part 11 - Limit Superior and Limit Inferior

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

###### Content of the video:

00:00 Intro

00:20 Example

02:07 Improper accumulation value

03:34 Definition limit superior and limit inferior

04:29 Why do we use these names and notations?

06:45 Fact

08:24 Credits

#### Part 12 - Examples for Limit Superior and Limit Inferior

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

###### Content of the video:

00:00 Intro

00:37 Example

01:31 Properties of lim sup and lim inf

03:00 Properties for combining 2 sequences

04:53 Example summation of 2 sequences

06:00 Example product of 2 sequences

06:57 Credits

#### Part 13 - Open, Closed and Compact Sets

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.

###### Content of the video:

00:00 Intro

00:14 Recalling (epsilon-)neighbourhoods

01:33 Example: neighbourhoods

02:43 Definition open sets

04:00 Definition closed set

04:43 Examples

06:11 Criterion for checking closeness with the help of sequences

06:44 Example for the criterion

07:40 Definition compact sets

08:58 Credits

#### Part 14 - Heine-Borel Theorem

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:

###### Content of the video:

0:00 Introduction

0:20 Sequentially compact set

1:07 Empty set is compact

1:17 {5} is compact

1:50 R is not compact

2:10 Proof: [a,b] is compact

3:15 Heine-Borel theorem statement

3:58 Proof of Heine-Borel theorem

#### Part 15 - Series - Introduction

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.

###### Content of the video:

00:00 Intro

00:19 Introducing series

01:07 Example of a series

02:46 Definition series

04:18 Rewriting the previous example

05:04 Another example

05:48 Credits

#### Part 16 - Geometric Series and Harmonic Series

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

###### Content of the video:

00:00 Intro

00:14 Recalling series

00:42 Geometric series

04:05 Harmonic series

09:05 Credits

#### Part 17 - Cauchy Criterion

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**:

###### Content of the video:

00:00 Intro

00:36 Properties of series

02:20 Cauchy Criterion

03:46 Proof

05:29 Example

07:15 Conclusion

08:47 Credits

#### Part 18 - Leibniz Criterion

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

###### Content of the video:

00:00 Intro

00:18 Recalling the harmonic series

01:24 Leibniz Criterion

03:07 Proof

07:52 Example

08:35 Credits

#### Part 19 - Comparison Test

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.

###### Content of the video:

00:00 Intro

00:21 Absolutely convergent

00:47 Absolutely convergent implies convergent

02:06 Implication doesn’t work in the other direction

02:45 Majorant criterion

04:16 Proof of majorant criterion

05:31 Minorant criterion

06:30 Example for the minorant criterion

07:44 Credits

#### Part 20 - Ratio and Root Test

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

###### Content of the video:

00:00 Intro

00:50 Recalling the geometric series

01:26 Proposition

02:33 Ratio test

03:43 Proof of the ratio test

05:14 Example using the ratio test

07:01 Root test

07:37 Proof of the root test

08:21 Example using the root test

09:15 Root criterion (limit form)

10:58 Credits

#### Part 21 - Reordering for Series

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:

###### Content of the video:

00:00 Intro

00:17 Meaning of reordering of series

01:13 Example: Reordering can change the value

02:32 2nd Example with a convergent series

04:58 Definition Reordering

06:10 Theorem for abs. convergent series

06:39 Proof of the Theorem

12:22 Credits

#### Part 22 - Cauchy Product

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

###### Content of the video:

00:00 Intro

00:38 Looking at finite sums

02:30 Definition Cauchy Product

04:07 Theorem about abs. convergence

04:49 Example for the theorem

07:51 Credits

#### Part 23 - Sequence of Functions

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

###### Content of the video:

00:00 Intro

00:19 Definition function

01:06 Continuous function

02:20 Definition bounded function

03:36 Definition sequence of functions

05:57 Credits

#### Part 24 - Pointwise Convergence

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

###### Content of the video:

00:00 Intro

00:33 Pointwise convergence

02:03 1st Example

03:20 2nd Example

06:40 3rd Example

07:43 Credits

#### Part 25 - Uniform Convergence

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

###### Content of the video:

00:00 Intro

00:44 Uniform convergence

02:52 Measuring distance between functions

04:30 Uniform convergence using supremum norm

05:06 Example

06:28 Uniform convergence is stronger than pointwise convergence

07:30 Credits

If you want to learn more how to show uniform convergence for special examples, you can check out the supplementary video about the Weierstrass M-Test.

#### Part 26 - Limits of Functionss

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

###### Content of the video:

00:00 Intro

00:59 Definition

05:50 1st Example

06:29 2nd Example (Polynomial)

08:12 Credits

#### Part 27 - Continuity and Examples

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**:

###### Content of the video:

00:00 Intro

00:18 Definition Continuity

01:22 Special case (holes in the domain of definition)

02:56 Extended definition of continuity

03:48 Examples

09:28 Credits

#### Part 28 - Epsilon-Delta Definition

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

###### Content of the video:

00:00 Intro

00:40 Epsilon-Delta criterion

04:32 Proof

08:28 Credits

#### Part 29 - Combination of Continuous Functions

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**.

###### Content of the video:

00:00 Intro

00:14 Recalling continuity

01:25 Combining 2 continuous functions

03:07 Composition of functions

05:55 Proof for composition of functions

07:27 Credits

#### Part 30 - Continuous Images of Compact Sets are Compact

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

###### Content of the video:

00:00 Intro

00:14 A special property of continuous functions

01:09 Theorem about images of compact sets

03:25 Proof of the Theorem

06:33 Credits

#### Part 31 - Uniform Limits of Continuous Functions are Continuous

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

###### Content of the video:

00:00 Intro

00:14 Uniform convergence for sequence of functions

01:09 Theorem for uniform limit of continuous functions

02:18 Proof of the Theorem

07:41 Credits

#### Part 32 - Intermediate Value Theorem

The **intermediate value theorem**, finally!

###### Content of the video:

00:00 Intro

00:14 Introducing the Intermediate Value Theorem

01:26 Definition Intermediate Value Theorem

02:29 Corollary

02:56 Proof of the Intermediate Value Theorem

08:16 Credits

#### Part 33 - Some Continuous Functions

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** and we denote it by $ \log $. Please note that this is the **natural logarithm** function which one often sees written as $ \ln $ as well. Moreover, in this video, we can generalise the whole concept and find so-called **power series**:

###### Content of the video:

00:00 Intro

00:25 Exponential Function

03:27 Logarithm Function

05:00 Polynomials

05:33 Power Series

08:49 Credits

#### Part 34 - Differentiability

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

#### Part 35 - Properties for Derivatives

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

#### Part 36 - Chain Rule

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

#### Part 37 - Uniform Convergence for 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.

#### Part 38 - Examples of Derivatives and Power Series

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

#### Part 39 -Derivatives of Inverse Functions

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:

#### Part 40 - Local Extreme and Rolle’s Theorem

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:

#### Part 41 - Mean Value Theorem

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

#### Part 42 - L’Hospital’s Rule

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

#### Part 43 - Other L’Hospital’s Rules

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

#### Part 44 - Higher Derivatives

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

#### Part 45 - Taylor’s Theorem

Now we finally discuss **Taylor’s Theorem**:

#### Part 46 - Application for Taylor’s Theorem

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

#### Part 47 - Proof of Taylor’s Theorem

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

#### Part 48 - Riemann Integral - Partitions

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

#### Part 49 - Riemann Integral for Step Functions

Now we can define the **Riemann integral for step functions**:

#### Part 50 - Properties of the Riemann Integral for Step Functions

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

#### Part 51 - Riemann Integral - Definition

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

#### Part 52 - Riemann Integral - Examples

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

#### Part 53 - Riemann Integral - Properties

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

#### Part 54 - First Fundamental Theorem of Calculus

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

#### Part 55 - Second Fundamental Theorem of Calculus

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

#### Part 56 - Proof of the Fundamental Theorem of Calculus

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

#### Part 57 - Integration by Substitution

Next, we will discuss some integration rules. We start with **substitution**.

#### Part 58 - Integration by Parts

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.

#### Part 59 - Integration by Partial Fraction Decomposition

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.

#### Part 60 - Integrals on Unbounded Domains

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.

#### Part 61 - Comparison Test for Integrals

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:

#### Part 62 - Integral Test for Series

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:

#### Part 63 - Improper Riemann Integrals for Unbounded Domains

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**:

#### Part 64 - Cauchy Principal Value

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**:

#### Connections to other courses

#### Summary of the course Real Analysis

- You can download the whole PDF here and the whole dark PDF.
- You can download the whole printable PDF here.
- Test your knowledge in a full quiz.