Lecture notes of Mathematical analysis

Lecture notes of Mathematical analysis

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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

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