Here, you find my whole video series about Start Learning Reals in the correct order and I also help you with some text around the videos. If you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:
Start Learning Reals is a video series I started for everyone who is interested in the necessary foundations you need to enjoy the whole world of mathematics. Now we are ready to start talking about the real numbers, which are so important in all calculus:
Let’s get started
The term Cauchy sequence, you now have learnt, is very important in your mathematical career. Such a sequence describes the notion that we get closer and closer to a point. Looking at such Cauchy sequences of rational numbers, we find the first problem:
We have seen that the to notions Cauchy sequence and convergene sequence are not equivalent in the rational numbers. This is indeed a heavy disadvantage, because it means that we can get closer and closer to a point of the number line by using rational numbers but this point is not a rational number anymore. The real numbers will fix this problem. However, before we construct them let us use the axiomatic method to calculate with them first.