Category:

# Basic and Real Analysis

Enrolled: 1 student
Level: Intermediate

Real Analysis is a rigorous and foundational course in advanced mathematics that delves into the study of real numbers, functions, and their properties. Real Analysis provides the tools and techniques necessary for a deeper understanding of calculus, as well as for the exploration of more advanced mathematical subjects. This course aims to equip students with a solid theoretical foundation and problem-solving skills essential for further studies in mathematics, physics, engineering, and related disciplines.

Course Objectives:

1. Develop a rigorous understanding of the real number system, including properties of real numbers, order relations, and completeness.
2. Study the concepts of limits, continuity, and differentiability of functions, and explore their applications in calculus.
3. Investigate the convergence and divergence of sequences and series, and analyze their properties.
4. Introduce the theory of integration, including Riemann integration and its fundamental properties.
5. Explore the fundamental concepts of metric spaces and topological spaces, and their relevance in analysis.
6. Examine the properties of continuous functions and their implications on connectedness and compactness.
7. Introduce the concept of differentiation in higher dimensions, including partial derivatives and the chain rule.
8. Study the theory of infinite series and power series expansions, and analyze their convergence and differentiability.
9. Investigate the fundamental theorems of calculus, such as the mean value theorem and the fundamental theorem of calculus.
10. Develop proficiency in constructing rigorous mathematical proofs and logical reasoning.

### Sequences

1
Introduction
4:23
2
Sequences and limits
3
Bounded sequences and unique limits
9:34
4
Theorem on limits
6:38
5
Sandwich theorem
8:18
6
Supremum and Infimum
9:09
7
Cauchy sequences and Completeness
9:00
8
Example Calculation
10:02
9
Subsequences and accumulation values
8:04
10
Bolzano-Weierstrass theorem
5:57
11
Limit superior and limit inferior
8:53
12
Examples for Limit superior and limit inferior
7:27
13
Open, Closed and Compact Sets
14
Heine-Borel theorem

### Series

1
Series - Introduction
6:03
2
Geometric Series and Harmonic Series
9:36
3
Cauchy Criterion
9:16
4
Leibniz Criterion
9:04
5
Comparison Test
8:15
6
Ratio and Root Test
11:14
7
Reordering for Series
12:38
8
Cauchy Product
8:22

### Functions

1
Sequence of Functions
6:13
2
Pointwise Convergence
8:13
3
Uniform Convergence
7:47
4
Limits for Functions
8:29
5
Continuity and Examples
9:45
6
Epsilon-Delta Definition
8:58
7
Combination of Continuous Functions
7:55
8
Continuous Images of Compact Sets are Compact
7:03
9
Uniform Limits of Continuous Functions are Continuous
8:11
10
Intermediate Value Theorem
8:45
11
Some Continuous Functions
9:19

### Differentiation

1
Differentiability
10:49
2
Properties for Derivatives
10:10
3
Chain Rule
6:16
4
Uniform Convergence for Differentiable Functions
7:25
5
Examples of Derivatives and Power Series
12:37
6
Derivatives of Inverse Functions
8:34
7
Local Extrema and Rolle's Theorem
11:38
8
Mean Value Theorem
7:40
9
L'Hôpital's Rule
11:16
10
Other L'Hôpital's Rules
13:14
11
Higher Derivatives
13:00
12
Taylor's Theorem
10:15
13
Application for Taylor's Theorem
9:35
14
Proof of Taylor's Theorem
10:55

### Integration

1
Riemann Integral - Partitions
10:44
2
Riemann Integral for Step Functions
10:00
3
Properties of the Riemann Integral for Step Functions
9:25
4
Riemann Integral - Definition
5:38
5
Riemann Integral - Examples
12:59
6
Riemann Integral - Properties
7:44
7
First Fundamental Theorem of Calculus
10:21
8
Second Fundamental Theorem of Calculus
7:37
9
Proof of the Fundamental Theorem of Calculus
12:47
10
Integration by Substitution
16:16
11
Integration by Parts
8:02
12
Integration by Partial Fraction Decomposition
15:47
13
Integrals on Unbounded Domains
9:04
14
Comparison Test for Integrals
7:40
15
Integral Test for Series
11:56
16
Improper Riemann-Integrals for Unbounded Functions
7:59
17
Cauchy Principal Value
9:15

### Examination Questions

1
Problem Solving - Series and Convergent Criteria
1:29:48
2
Problem Solving - Continuous Functions
2:13:25
3
Problem Solving - Differentiable Functions
1:30:17