实数(the real number)
The so-called differentiable (可微） functions constitute the main object of study of classical analysis.
1 The Axiom System(公理体系） and some General Properties of the Set of Real Numbers
1.1实数集的定义
定义1 满足以下四组公理条件的集叫实数集，他的元素叫实数。
加法公理：
An operation
（the operation of addition）is defined,assigning(赋值) to each ordered pair （x,y）of element x,y of a certain element called the sum of x and y.
.There exists a neutral(中性的) ,or identity element 0 such that:
For every
.For every element there exists an element called the negetive of such that:
.The operation + is associative(结合的),that is,the relation:
Holds for any elements of.
.The operation + is commutative,that is,
For any elements of .
If an operation is defined on a set G satisfying axioms , ,,we say that a group structure is defined on G or that G is a group.condition holds,the group is called commutative or Abelian.So is an abelian group(阿贝尔群).
乘法公理：
An operation
(the operation of multiplication) is defined,assigning to each ordered pair of elements of a certain element called the product of and .This operation satisfies the following condition:
.There exists a neutral,or identify element such that:
For every
.For every element there\exists an element called the inverse(逆) or reciprocal(倒数) of ,such that:
The operation is associative,that is,the relation:
The operation is commutative,that is:
For any elements of .
The connection between addition and multiplication
Multiplication is distributive with respect to addition,that is
For all .
If two operations satisfying these axioms are defined on a set G,then G is called a field.
序公理
Between elements of there is a relation ,that is ,for elements one can determine whether or not. Here the conditions must hold:
. .
. .
.
The relation on is called inquality(不等关系).A set（集合）on which there is a relation between pairs of elements satisfying axioms ,,,as you know,is said to be partially ordered(偏序集).If addition axioms holds,the set is linearly ordered(线性序集) by the relation of inequality between elements.
If are elements of ,then
If and are elements of ,then
完备定理
If and are nonempty(非空) subsets(子集) of having the property that for every and every ,then there exists such that for all and .
CONSISTENCY(存在性) AND UNIQUENESS(唯一性) OF AXIOMS AHEAD
Uniqueness:If two people A and B construct models independently,say of number systems and ,satisfying the axioms,then a bijective（双射） correspondence can be established between the systems and ,say f:,preserving（保持） the arithmetic（算数） operation and the order,that is:
和 只是实数上的具体实例，这种实例叫做同构实现(isomorphic),映射f称为同构(isomorphism).The result of this mathematical activity is thus not about any particular realization,but about each model in the class of isomorphic models of the given axioms system.
1 Consequences of the Addition Axioms
.There is only one zero in the set of real numbers.
.Each element of the set of real numbers has a unique negative.
.In the set of real numbers the equation
Has the unique solution
2 Consequences of the Multiplication Axioms
. There is only one multiplicative unit in the real numbers.
.For each there is only one reciprocal .
.For ,the equation has the unique solution .
3 Consequences(推论) of Axioms Connecting Addition and Multiplication
. For any
.
.For any
.For any
. For any
4 Consequences of Order Axioms
.For any x and y inprecisely one of following relations holds:
.For any
,
5 Consequences of Axioms Connecting Order with Addition and Multiplication
.For any
.If ,then
.0<1.
.
The Completeness Axioms(完备公理) and the Existence of a Least Upper(or Greatest Lower) Bound of a Set of Numbers(上下确界的存在性)
Definition 2:
A set is said to be bounded above( resp.bound below) if there exists a number such that for all .
Definition 3:
A set that is bounded both above and below is called bounded.(既有上界又有下界的集合叫做有界集).
Definition 4:
An element is called the largest or maximal element of if for all .
Definition 5:
The smallest number that bounds a set from above is called the least upper bound of and denoted sup (read “the suppremum of ”) or .
Lemma.(the least upper bound principle)(确界原理)
Every nonempty(非空的) set of numbers is unique,we need only verify（证明） that the least upper bound exists.
Lemma.（X bounded below） inf X)
2.2 The Most Important Class of Real Numbers and Computational Aspects of Operations with Real Numbers(最重要的实数类和部分实数数值计算)
2.2.1 The Natural Numbers
Definition 1:
A set is inductive(归纳集) if for each number ,it also contains x+1.
Definition 2:
The set of natural numbers is the smallest inductive set containing 1, that is ,the intersection of all inductive sets that contain 1.(包含数1的最小的归纳集，即含数1的一切归纳集之交，叫自然数集。)（也可以从0开始）
The Principle of Mathematical Induction:
If a subset E of the set of natural numbers N is such that and together with each number , the number x+1 also belongs to E,then E=N.
. The sum and product of natural numbers are natural numbers.
.
.For any the set contains a minimal element,namely
.
.The number is the immediate successor of the number ;that is ,if ,there are no natural numbers x satisfying .
.If and ,then and (n-1) is the immediate predecessor of n in N;that is ,if there are no natural numbers x satisfying n-1<x<n.
.In any nonempty subset of the set of natural numbers there is a minimal element.
Rational(有理数) and irrational
Definition:The union of the set of natural numbers,the set of negatives of natural numbers,and zero is called the set of integers and is donated .
Addition and multiplication of integer numbers do not take us outside of .
Numbers of the form ,where ,are called rational.
In a certain sense nearly all real numbers are irrational.
Among the irrational numbers we make a further distinction between the so-called algebraic irrational numbers and the transcendental numbers.(无理数又分为代数无理数和超越数).
A real number is called algebraic if it is the root(根) of an algebraic equation（代数方程）
with rational cofficients.
Otherwise the number is called tranacendental.(超越数)
THE PRINCIPLE OF ARCHIMEDES
We remark that the propositions that we have proved up to now about the natural numbers and the integers have made no use at all of the completeness axiom.
.Any nonempty subset of natural numbers that is bounded from above contains a maximal element.(自然数集的任何不空有界集中有最大元)
Corollaries .The set of natural numbers is not bounded above.( 自然数集没有上界)
.Any nonempty subset of integers that is bounded from above contains a maximal element.
.The set of integers is unbounded above and unbounded below.
(The principle of Archimedes)
For any fixed(固定的) positive number h and any number x there exists a unique integer k such that.
Corollaries:
.For any positive number there exists a natural number n such that .
.If the number is such that and for all ,then x=0.
.For any numbers such that a<b there is a rational number such that a<r<b.
For any number there exists a unique integer such that .
的完备定理用几何语言说就是直线L上没有洞能把L分成没有公共点的两块（这样的分划只能用直线自己的点实现）。
Definition: An open interval containing the point will be called a neighborhood of this point.
In particular,when ,the open interval is called the -neighborhood of this point. Defining a number by successive approximations
Definition:if x is the exact value of a quantity and a known approximation to the quantity,the numbers:
and
are called respectively the absolute and relative error of approximation by .
If
Then
If,in addition,
Then
THE POSITIONAL COMPUTATION SYSTEM(位置计数法)
Lemma:If a number q>1 is fixed,then for every positive number there exists a unique integer such that:
Fixed q>1 and take an arbitrary positive number ,by the lemma we find a unique number such that .
Definition:The number p satisfying is called the order of x in the base q or simply the order of x.
We can constructed a sequence of rational numbers of the special form:
And such that
BASIC LEMMAS CONNECTED WITH THE COMPLETENESS OF THE REAL NUMBERS
THE NESTED INTERVAL LEMMA(Cauchy-Cantor Principle)(区间套定理)
Definition:A function of a natural-number argument is called a sequence or,more fully,a sequence of elements of .
Definition:Let be a sequence of sets.If that is for all ,we say the sequence is nested.
Lemma:For any nested sequence
of closed intervals,there exists a point belonging to all of these intervals.
If in addition it is known that for any there is an interval whose length is less than .then c is the unique point common to all the intervals.
THE FINITE COVERING LEMMA(BOREL-LEBESGUE PRINCIPLE,OR HEINE-BOREL THEPREM)
Definition:A system of sets X is said to cover a set Y if ,(that is,if every element belong to at least one of the set in the system S).
Lemma:Every system of open intervals covering a closed interval contains a finite subsystem that covers the closed interval.
THE LIMIT POINT LEMMA(BOLZANO-WEIERSTRASS PRINCIPLE)
Definition:A point is a limit point of the set if every neighborhood of the point contains an infinite subset of .
This condition is obviously equivalent to the assertion that every neighborhood of p contains at least one point of different from p itself.
Lemma:Every bounded(有界) infinite(无限) set of real numbers has at least one limit point.
COUNTABLE AND UNCOUNTABLE SETS
Definition:A set is countable if it is equipollent with the set N of natural numbers,that is,card =card .
A)An infinite subset of a countable set is countable.
B)The union of the sets of a finite or countable system of countable sets is a countable set.
It follows from the proposition just proved that any subset of a countable set is either finite or countable.
If it is known that a set is either finite or countable ,we say it is at most countable(至多可数集).(An equivalent expression is card card .)
Corollaries:
1) card =card .
2) card .
3) card Q=card ,that is,the set of rational numbers is countable.
4) The set of algebraic（代数的） numbers is countable.
The algebraic equations with rational coefficients(of arbitrary degree(任何范围))also form a countable set,being a countable union(over degree)of countable sets.
Definition:The set of real numbers is also called the number continuum(数的连续统),and its cardinality(势) the cardinality of the continuum.
Theorem:(Cantor)card N<card .
Corollaries:
1),and so irrational numbers exist.
2)There exist transcendental(超越数)numbers,since the set of algebraic numbers is countable.
LIMIT
Definition:A function whose domain(定义域) of definition is the set of natural numbers is called a sequence.
Throughout the next few section we shall be considering only sequences of real numbers.
Definition:A number is called the limit of the numerical sequence if for every neighborhood V(A) of A there exists an index N(depending on V(A) such that all terms of the sequence having index larger than N belong to the neighborhood V（A）.
Definition:If there exists a number A and an index N such that for all n>N,the sequence will be called ultimately constant.
Definition:A sequence is bounded if there exists M such that for all .
Theorem:
A)An ultimately constant sequence converges.
B)Any neighborhood of the limit of a sequence contains all but a finite number of terms of the sequence.
C)A convergent sequence cannot have two different limits.
D)A convergent sequence is bounded.
Theorem:Let and be numerical sequence.If and ,then
A)
B)
C)provided and .
Theorem:
A)Let and be two convergent sequences with and .If A<B,then there exists an index such that for all n>N.
B)Suppose the sequences ,,and are such that for all .If the sequence and both converge to the same limit,then the sequence also converges to that limit.
Corollary:Suppose and .If there exists N such that for all we have:
A)
B)
C)
D)
It is worth noting that strict inequality(不等式) may become equality in the limit.For example for all yet .
The Cauchy Criterion
Definition:A sequence is called a funsamental or Cauchy sequence if for any there exists an index such that whenever and .
Theorem:(Cauchy’s convergence criterion) A numerical sequence(数列) converges if and only if it is a Cauchy sequence.
Definition:A sequence is increasing if for all ,nondecreasing if for all ,Sequences of these four types are called monotonic(单调) sequences.
Definition:A sequence is bounded above if there exists a number M such that for all .
Theorem:(Weierstrass)In order for a nondecreasing sequence to have a limit it is necessary and sufficient(充分的) that it be bounded above.
Definition:
Definition:If is a sequence and an increasing sequence of natural numbers,then the sequence is called a subsequence of this sequence.
Lemma:(Bolzano-Weierstrass)Every bounded sequence of real numbers contains a convergent subsequence.
Lemma:From each sequence of real numbers one can extract either a convergent subsequence or a subsequence that tends to infinity.
Definition:The number iscalled the inferior limit of the sequence and denoted or .If ,it is said that the inferior limit of the sequence equals positive infinity,and we write or .
Definition:A number is called a partial limit of a sequence ,if the sequence contains a subsequence converging to that number.
Proposition 1:
The inferior and superior limits of a bounded sequence are respectively the smallest and largest partial limits of the sequence.
Proposition 2 :For any sequence,the inferior limit is the smallest of its partial limits and the superior limit is the largest of its partial limits.
Corollary:A sequence has a limit or tends to negative or positive infinity if and only if its inferior and superior limits are the same.
Corollary:A sequence converges if and only if every subsequence of it converges.
Corollary:The Bolzano-Weierstrass Lemma in its restricted and wider formulations follows from Propositions 1 and 2 respectively.
ELEMENTARY FACTS ABLOT SERIES
Definition:The expression is denoted by the symbol and usually called a series or an infinite series.
Definition:The elements of the sequence ,when regarded as elements of the series,are called the terms of the series.the element is called the nth term.
Definition:The sum is called the partial sum of the series,or,when one wishes to exhibit its index,the nth partial sum of the series.
Definition:If the sequence of partial sums a series converges,we say the series is convergent.If the sequence does not have a limit,we say the series is divergent.
Definition: The limit of partial sums of the series,if it exists,is called the sum of the series.
Theorem:(The Cauchy convergence criterion for a series) The series converges if and only if for every there exists such that the inequalities imply .
Corollary:If only a finite number of terms of a series are changed,the resulting new series will converge if the original series did and diverge if it diverged.
Corollary:A necessary condition for convergence of the series is that the terms tend to zero as ,that is,it is necessary that .
Definition:The series converges is absolutely convergent if the series converges.
Theorem:(Criterion for convergence of series of nonnegative terms) A series whose terms are nonnegative converges if and only if the sequence of partial sums is bounded above.
Theorem:(Comparison theorem)Let and be two series with nonnegative terms.If there exists an index N such that for all ,then the convergence of the series implies the convergence of ,and the divergence of implies the divergence of .
Corollary:(The Weierstrass M-test for absolute convergence)Let and be series.Suppose there exists an index N such that for all ,Then a sufficient condition for absolute convergence of the series is that the series
Converge.
Corollary:(Cauchy’s test)Let be a given series and .Then the following are true:
A) if ,the series converges absolutely;
B) If ,the series diverges.
C) There exist both absolutely convergent and divergent series for which
Corollary:(d’Alembert’s test)Suppose the limit exists for the series .then,
A)if ,the series converges absolutely.
B)If ,the series diverges.
C)There exists both absolutely convergent and divergent series for which .
Proposition(Cauchy)If ,the series converges if and only if the series converges.
Corollary:The series converges for and diverges for p.
THE LIMIT OF A FUNCTION
If for every there exists such that for every such that .
Definition:A deleted neighborhood of a point is a neighborhood of the point from which the point itself has been removed.
If a is a limit point of E,then for every neighborhood U(a).
Proposition:The relation holds if and only if for every sequence of points converging to a,the sequence converges to A.
We call the reader’s attention to the fact that,in order to establish the properties of the limit of a function,we need only two properties of deleted neighborhoods of a limit point of a set:
A) .
B) (任何两个去心邻域之交还是去心邻域)
A function is said to be infinitesimal(无穷小) as if .
THE GENERAL DEFINITION OF THE LIMIT OF A FUNCTION(limit over a Base)
BASES(基):DEFINITION AND ELEMENTARY PROPERTIES