Lecture 1 - Overview of the Course

Real Analysis is an analysis of $\mathbb{R}$. But it's more than that. It's the an analysis of the functions on it (namely $f:\mathbb{R}\to \mathbb{R}$). But first we have to know what $\mathbb{R}$ is, compared to something like $\mathbb{Q}$ or especially $\mathbb{N}$.

But first let's start with something more fundamental. Let's start with:

as a given. We know all the basic properties:

• addition
• multiplication
• commutative property
• etc.

This extends to the natural extension to the integers $\mathbb{Z}$:

Then when you want to extend these even further, you get the rationals $\mathbb{Q}$:

There's already some really nice properties of $\mathbb{Q}$. One thing is that $\mathbb{Q}$ forms a Field. Furthermore it has Order. These two properties together make it be an Ordered Field.

One interesting property of $\mathbb{Q}$ is that you can always find an element between two rational numbers. Take $p,q\in \mathbb{Q}$ where $p<q$ (by $\mathbb{Q}$ being ordered). Then choose $\frac{p+q}{2}$ as the in-between element:

But there are some lengths as we know that are irrational, such as $\sqrt{2}$. There are no $p,q\in \mathbb{Z}$ where $\frac{p}{q}=\sqrt{2}$. See the proof of it here.

This clearly shows that there are gaps in the rationals $\mathbb{Q}$. How do we fill these gaps such that we get all of them. The idea is that $\mathbb{R}$ should fill these gaps.

TODO

• #todo/school : MATH412 read section 1.1 📅 2024-09-24 ✅ 2024-09-23
• #todo/school : MATH412 do problems 1.2.1, 1.2.2 (no need to read 1.2) 📅 2024-09-24 ✅ 2024-09-23