The following session covers the questions that are most commonly asked when it comes to proofs. It finds out whether all mathematical statements are clearly true or clearly false using examples and calculations. Also, the professor tells you why you need to use proofs at all. You will understand how to know when something is completely false or true and when it is a special case. This is seen when trying to join two parts of a graph. You will also understand why you must know what your definitions are specifically. You do not need to memorize proofs, but you will need to understand how to do them and figure out the unusual ideas and tricks that you can use.

## Lecture 3: Math. Analysis – Bounded sets

This is a continuation of the previous class on opening closed balls in space in cases where the open balls are excluded and those points that were distant and are exactly the same radius from the center. It is less strict and includes are the dimensions on your ball’s circle. It covers the difference between the norm and the modulus. You can never use single line modulus for sectors. It also covers a discussion of bounded and unbounded sets by using examples. We also use a diagram to find out why the results are true and finding proofs that support our findings. You will understand what it means for a set to be bounded.

## Lecture 4a: Math. Analysis – Examples of bounded and unbounded d-cells

This video covers both two and three dimensions and teaches how to draw pictures. There are different scenarios created for different products. With two intervals, you get a triangle figure. If you get products it is easy to know which edges are missing and how to calculate and fill them up. You will also figure out which ones are included. The video concludes with the discussion of unbound b-cells. You need to remove the infinity because they are not real numbers, and the calculations need you to work with visible numbers. You also need to use non-negative real numbers it will make your calculations simpler.

## Lecture 4b: Math. Analysis – Bounded and unbounded d-cells continued

This is chapter 3, and it covers a more advanced topic on open subsets. It will discuss what it means to be open or closed. You will understand better subjects that were in the past only studies in generality. You will also learn what a metric is and why it is important in your calculation of distances. You will also define open and prove why open balls are open. Each notion of distance comes with its own notion of the open and closed ball. So when you change the notion of distance, you have to change the notion of the open and closed ball as well. This is known as analytic topology.

## Workshop 3: Math. Analysis – Examples Class 1

This is the first examples class, and it helps you learn what it means for the sets to be bounded. It will also help you know what it means to be unbounded. With these examples, you will learn how to justify your answers during the exams. You will also find out which statements are true and which ones are false. This video covers the solutions to about 6 questions and allows you to find out which sets can work together and which ones are not compatible. The professor suggests that you pause the videos, do the calculations on your own and find out how they are solved and whether or not you followed the right procedures.