Lecture notes of Mathematical analysis
Preliminary theory
Teaching purpose:
Mathematical analysis is a watershed between high school and university. Its core is the thought of limit. Before introducing the concept of limit, let's first review the knowledge of set theory and logic, so as to introduce the concept of real number, then we may explain the concept of mapping and the definition of unary real value function in the real number field. Finally, we may introduce the concept of a function of one variable in real number field. and we can also put forward the concept of sequence. In this way,we have laid a solid foundation for analysis.
Teaching programmes:
① Set and its operation
Definition of set
Definition of elements
The universal set and subset
Equivalence of set.
Intersection, Union and complement of set
The rules of set operation
② Logic
Proposition
Operations of proposition
Universal proposition and subalternate
(Duality principle)
③ Real number
Positive Integer (Mathematical Induction)
Integer
Rational number
Irrational number
Real number
Basic properties of Real Number
Axiom of continuity of real number
Supremum theorem
Closed interval theorem
④ Mapping and function
Definition of mapping
Interval, Ray and Neighborhood
Definition of function
Univariate real value function
⑤ Numerical sequence
Definition of numerical sequence
Common numerical sequence