From analaysis to Calculus

This is an easy course from a book - From Analysis to Calculus. I collect some essential stuff in it and discuss them.  
Before discussing them, we must define some related symbols and notations.

Real number Notations:
Given any a real number x , we can discuss its representation as below.
Assume that there exist a real number x and its range locates between 1 and 10 (not include two endpoints )  , we can list it as below. 
1 < x < 10  and the other representation as below:
x = { x ∈ R |  1 < x < 10 } , R is a real numbers set. Because real number has "稠密性" , so we can't list all numbers between them. In other words , if we can reduce this property , we should list all numbers that satisfy it. Basically , if we limit its numbers are integral Z , we can list them all. Again , we see an example. This example which just lists its solutions . It is very straight concept , e.g.  3X = 1 , X = 1/3 and it can be represented a set S = { x ∈ R | x = 1/3 }
okay, the above is some easy example that discuss some real numbers representations.