 # Part 4: Math. Analysis – Absorption of sequences by sets

Here, you will understand the generalizations of the sequences of Rd. you will also learn the absorption of sequences by sets. For this, all terms of the sequence have to be stuck inside a sequence. All integers of a sequence must be stuck in the set. You will also have to prove whether the sequences you are using are converging. You will also learn the Sandwich theorem and understand how you can generalize this theorem. You will also learn how to do algebra limits because these do not need proof. You will also learn about the edge of limits and limit holds.

## Workshop 5: Math. Analysis – Examples Class 2

When it comes to problem solving, nothing can guide you better in understanding the theorems than having examples to follow. This video contains about four examples that show you how to calculate the open sets and justify your proofs. It will help you understand and have better knowledge of sets and interior points within the sets. You need to do sketches that will help you understand the problem presented and find where the answer are. The sketches are sufficient justifications for two dimensions and other issues you may be presented with. Here, the professor teaches you how to create efficient sketches.

## Lecture 10a: Math. Analysis – Proof of the sequence criterion for closedness

This is the conclusion of the fifth chapter. It concludes by discussing the characterization of the closedness of subsets in terms of convergent sequences. When you take the sequence of a set, you may realize that its limitations-if any- are outside the set. This video has examples that will help you understand how the sequences work and why the sequences and sets are related and how their closedness affects your ability to find proof of sequence criterions. You will also figure out how to find complements. You will also see how diagrams will make your work easier during the exams and while completing a coursework.

## Lecture 10b: Math. Analysis – Subsequences and Sequential Compactness

Chapter six covers subsequences and sequential compactness which is more advanced. Even when sequences do not converge, you can take subsequences and find parts that do converge. You can converge certain subsequences and have the sequence converge. With examples, the professor explains what you need to understand to ensure that these points are driven home.  You will also learn more about empty, bounded closed cells and how they can be used to create subsequences that can converge.

## How do we do proofs? – Dr Joel Feinstein

In this session, the discussions are mainly about sequences. If the sequence is infinite, then the calculations are simpler. Different authors disagree on whether or not 0 is a natural number. Therefore you will have to bring this up with your professor and find out what they think. For this module, however, we assume that 0 is not a natural number. There are also calculations within this video that help you understand how you can apply these facts within the exams situation and in real life. You will also understand more about how to justify your answers formally whenever you are required to do so.