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