Download 1.1 real numbers

微积分与数学分析
Calculus and Elementary
Analysis (CEA)
主讲：
助教：
游雄
魏敏
2007-2009
NJAU mathematical analysis
Mathematical Analysis
Calculus
and
Elementary Analysis
(CEA)
Instructor: Xiong You
Teaching Assistant: Min Wei
2007.9-2009.1
Textbook:
1. East China Normal Uiv.:
Mathematical analysis. 3rd edition. Higher
Educational Press, 2002.
(In Chinese)
2. Finney, Weir and Giordano, Calculus 11th edition, Pearson
Education,2005.
3. M.H. Protter, Basic Elementary Real Analysis, Springer, 2004.
Reference:
1. W. Rudin: Principles of
mathematical analysis. McGraw Hill
Inc., 1976.
Grade Policy
Homework assignment: 20%
Classroom performance: 10%
Final exam: 70%
NJAU mathematical analysis
Chapter1
Real Numbers
and Real Functions
§1.1
§1.2
Least§1.3
§1.4
Real Numbers
Sets of Numbers,
the
RealUpper-Bound
Functions Principle
Composite Functions and Inverse
Functions
§1.1 Real
Numbers
a. Real numbers and their properties
b. Absolute values and inequalites
a. Real numbers and their properties
[Recall] Rational and Irrational Numbers
p

fractions
(positive,
negative,
zero)
 rational:
q

real 
( finite or infinite cycling decimals)
irrational: ( infinite noncycling decimals)


Notation for Sets of numbers
R ----real numbers
Q ----rational numbers
Z ----integers
N ----natural numbers
￭ Write Real Numbers in form of decimals
1) Irrational
a0 .a1a2
an
(infinite noncycling)
2) Rational
a0 .a1a2
ar c1c2
ck c1c2
ck
(infinite cycling)
or
a0 .a1a2
an  a0 .a1a2
(an  1)99 9
￭
Comparison of Real Numbers
1) Definition 1
Given two nonnegative real numbers,
x  a0 .a1a2 an , y  b0 .b1b2 bn ,
where a0 , b0
are nonnegative integers, ai , bi are
integers with
0  ai , bi  9, i  1, 2,
1) x=y , if
ai  bi , i  0,1, 2,
. We say that
;
2) x>y , if a0  b0 , or
a0  b0 , ai  bi , i  1, 2,
, k , but ak 1  bk 1.
￭
Comparison of Real Numbers
Convention:
Nonnegative numbers > negative numbers
For negative numbers,
1) x=y, if -x= -y ;
2) x>y , if -y= -x .
Definition 2 Let
x  a0 .a1a2 an  be
a nonnegative number.
1) The rational number x n
 a0 .a1a2 an
is called the n th lower approximation of x ;
2) The rational number
xn  xn
1

n
10
is called the n th upper approximation of x .
For a negative
we define
and
x  a0 .a1a2
xn   a0 .a1a2
an
,
1
an 
,
n
10
xn  a0 .a1a2
an .
Note that for any real number x,
x0  x1  x2  
,
x0  x1  x2  
PROPOPSITION 1 For real numbers
x  a0 .a1a2
,
y  b0 .b1b2
,
x  y if and only if
there is an integer n , such that xn  yn .
PROPERTIES OF REAL NUMBERS(1)
1. The set of real numbers is
closed under addition,
subtraction, multiplication and
division (with nonzero denominator),
that is, the sum, difference, product or
quotient (with nonzero denominator) of
two real numbers is a real number.
PROPERTIES OF REAL NUMBERS(2)
2. The set of real numbers is
ordered , that is, for two real numbers
a and b, exactly one of the following
relations is true:
a < b, a = b, or a > b .
PROPERTIES OF REAL NUMBERS(3)
3. The order of real numbers
is transitive , that is,
if
a<b and b<c,
then
a < c.
PROPERTIES OF REAL NUMBERS(4)
4. Real numbers has the
Archimedes Property, that is,
for any real number b and a>0, there is a
natural number n
such that
b < na.
PROPERTIES OF REAL NUMBERS(5)
5. The real number set R is
dense, that is, between any two real
number a < b, there are other real
numbers, both rational and irrational.
PROPERTIES OF REAL
NUMBERS(6)
6. The real number set R are
one-to-one correspondent
to the set of points of the
real line.
x
。
Example 1 Let x and y be real numbers. Show that
there is a rational number r, such that x < r < y.
Proof.
Since x<y, there a nonnegative integer n, such
that x n  yn . Let
rational, and


r  x n  yn / 2.
x  xn  r  yn  y.
Then r is
Example 2 Let a and b be real numbers. Show that
If for any positive ε, a < b+ ε , then
a  b.
Proof. (Proof by contradiction) Suppose that a>b and
let ε = a-b. Then ε >0 and a = b + ε . This contradicts the
assumption. Therefore, it must be true that a  b.
b. Absolute values and
inequalities
DEFINITION The absolute value |x| of a
real number x is defined by
 a , a  0,
| a | 
a , a  0.
从数轴上看的绝对值就是到原点的距离：
-a
0
a
Some properties of absolute values
1. | a |  | a |  0 ; and | a |  0 only when a  0.
2 . - | a|  a | a| .
3. | a|  h  -h < a < h ; | a |  h  h  a  h , h  0.
4. a  b  a  b  a  b .
5. | ab || a || b |.
6.
a |a|

, b  0.
b |b|
QUESTION
Let a and b be real numbers.
If for any positive ε, |a – b|< ε , what
order is between a and b?
Some important inequalities(1):
1 Mean value inequalities.
 0
a  b  2ab (a, b  R).
2
2
and the equality holds

.
(算术平均值)
(几何平均值)
a  b  c  3abc (a, b, c  0).
3
3
3
(调和平均值)
Some important inequalities(1)
1 Mean value inequalities.
1
a  b  2 ab (a, b  0).
(算术平均值)
a  b  c  3 abc (a, b, c  0).
3
or
(几何平均值)
ab
 2 ab 
 a 2  b 2 (a, b  0).
1 1
2
(调和平均值)

a b
2

for
a
,
a
,
,
a

R
, we have,
In general,
1 2
n
H  G  A  Q,
n
where
H
⑵ 均值不等式:
1 1
 
a1 a2
G
1

an

1
n
1 1

n i 1 ai

n
n
,
1

i(算术平均值)
1 ai
1
n
n
a1a2
a1  a2 
A
n
an
 an
 n

   ai  ,
 i 1

1 n
  ai ,
n i 1
n
Q
a  a2 
n
2
1
2
 an
2

a
i 1
n
2
i
.
Some important inequalities(2):
2 Cauchy inequalities.
ac  bd  a2  b2  c2  d 2 (a, b, c, d  R).
(算术平均值)
In general,
n
a b
i 1
i i
and the equality holds

n
n
 a b
2
i 1

i
i 1
i
2
.
(调和平均值)
.
Some important inequalities(3):
有平均值不等式:
3 Bernoulli inequalities.
等号当且仅当
⑶ Bernoulli
不等式:
For
x  1, x 
0, n  N , n  2,
时成立.
(1  h) n  1  nh.
Proof. (Proof by induction)
If x  1  0,1  x  1, n  N and n  2,
(1  x)  n 1  (1  x)  1  1   1
n
n
 n n (1  x ) n  n (1  x ).
 (1  x)  1  nx.
n
For h  0, since
⑷ 利用二项展开式得到的不等式:
n(n  1) 2 n(n  1)( n  2) 3
n
(1  h)  1  nh  由二项展开式
h 
h    hn ,
2!
3!
More useful:
then
(1  h)  any term, or
the sum of any parts of r.h.s.
n
For h  0, since
⑷ 利用二项展开式得到的不等式:
n(n  1) 2 n(n  1)( n  2) 3
n
(1  h)  1  nh  由二项展开式
h 
h    hn ,
2!
3!
Question: If
then
(1  h)  any term, or
the sum of any parts of r.h.s.
n
(right hand side).
Some important inequalities(4):
4 Other useful inequalities.
sin x
 1,
(算术平均值)
sin x

x .
For 0  x   / 2,
0  sin x  x  tan x.
(几何平均值)
Homework
Today
P4. 3, 4, 6, 7
Class is over. See you next time!