References:
- Fundamentals of Abstract Analysis (Andrew M. Gleason)
- Understanding Analysis (Stephen Abbott)
- Principles of Mathematical Analysis (Walter Rudin)
As the idea of this website is to keep it simple, let me try to explain in plain words why do we start studying Real Analysis with the Real Numbers system.
The idea is to show that we start from the most simple number system (the Natural Numbers) and we reach an understanding of what is the so-called Real Numbers (another number system). In this process, we are interested in showing that the Real Numbers are a Complete Ordered Field. So, like math always loves to do, we need to define the words Complete, Ordered, and Field.
What is a Field?
A field is a set. Any set has its properties. In this set, the properties are: you can do two operations, addition and multiplication. Therefore, if you take any two elements of this set, you can add them, subtract them (which is the same as adding an additive inverse) and multiply them, and divide them (which is the same as multiply a number by it’s inverse, keeping the denominator different than zero).
Thus, any set you take that has these properties will be called a Field.
What is Order?
Order is related to the idea of who dominates who. Also, the idea of relating objects to each other is also implied here. Thus, let define what relation is.
Relation: A relation is a set. Any set has its properties. In this set, the properties are: an object is related to another object (or multiple objects are being related to each other at the same time). We represent a relation with a symbol written between the objects being related.
If a and b are being related to each other by a relation R, we say: aRb (as if we were talking aboue a < b, where R would stand for ‘>’).
Since we know that R is a set, it is composes by its elements. In this case, elements of R are called ordered pairs (a,b). Let’s define what ordered pairs are.
Relation > Ordered Pairs: This can be tricky, but let’s keep it simple. Ordered pairs are mathematical objects in which its elements have a well-defined order in which they are presented visually. As mentioned, we represent them as (a,b), so (b,a) is different from the former. Also, notice that ‘(a,b)‘ itself is an object, but we said that they are elements of a set R. Let’s first focus on the idea of (a,b) representing elements of a set. What is this set in a broader sense? It’s the set generated by the cartesian product of two other sets. Let’s call them A and B. What is cartesian product? Let’s define it.
Relation > Ordered Paris > Cartesian Product: 
ii) Definition of Binary Operations: A binary operation in the set S is basically a function from 
(Very) Intuitive Mathematics 