Вступ до аналізу. Ч. 1

" , * " " ", " ( ! ", " ( -

+ , * " , ( 0) " " ( . 5).

$ . 5.

# ! * " ( x , * -

, , + ( , *

( ( . 6).

$ . 6.

# + + ' ( + '

, *. # * -

“( x ” ! + “ x ”.

. ( a b – ( , a < b .

# ( ) [a, b] , * " + x

, * ", * a ≤ x ≤ b .

$ (a, b ) , * " + x , -

* ", * a < x < b .

( ) [a, b ) (a, b] , * "

+ x , * ", * a ≤ x < b a < x ≤ b .

, ' ' [a, b] (a, b ) , ,

a, b + *, + *. & -

( + *, 2 .

0 , , + * ! : (−∞, b), (a, + ∞), (−∞, b], [a, + ∞), (−∞, + ∞) . 0 (−∞, + ∞) + '-

, * " ,' + ' ( , ! = (−∞, + ∞) .

. ( x0 – * ( . % x0 , * " ! * " ( (α, β), " ( * x0 , ! α < x0 < β .

12

., (1, 2), (−∞, 3), (−∞, + ∞) , x0 = 1, 7 . 6 +, x0 , +, !’" ! -

(α, β) ( . 7).

	$ . 7.
6 ( δ > 0 .
δ– x0	, * " ( x0 − δ, x0 + δ) .
# + x0	, * . ? δ ' * -

( . 8).

$ . 8.

. ( ( x .

( ) x ' * | x | , " -

, * " ':

x,

&

x ≥ 0,

| x |=

−x,

&

x < 0.

.:

5

= 5,

−3

= 3,

0

= 0,

3 − π

= π − 3 .

6 | x |

', x

( . 9).

$ . 9.

13

', (! *2 + ": x, y :

1)x ≥ 0 .

2)x = 0 * , x = 0 .

3)−| x | ≤ x ≤ | x |.

4)− x = x .

5)| xy | = | x | | y | .

6)	x	=			x			( y ≠ 0) .

yy

7)x + y ≤ x + y .

8)x − y ≥ x − y .

9)| x | − | y | ≤ | x − y | .

. ( x, y – . & * + , , - ', x − y ( . 10).

$ . 10.

$" δ – x0 , x – * *

( . 11).

$ . 11.

14

# , x − x0 < δ . , ! "

x − x0 < δ , x ( x0 − δ, x0 + δ) . 6, " x < a , a – "

, x (−a, a ) .

& ' ( x0 − δ, x0 + δ) x0 , ! " - + ( x0 − δ, x0 ) ( x0 , x0 + δ) . # , (

x0 (! ). " ! * " x * :

0 < x − x0 < δ .

. ' ( x , * " (! *2 -, " ! *2, + x .

4 x , * " " [x] ! entier( x) . . :

[3, 7] = 3;				= 1; [5] = 5; [−7,1] = −8 .
		2

? x * ", ( *: [x] ≤ x < [x] +1.

. ( x , * " "

+ x ( ' '.

% , * " ! x " {x}. # : {x} = x − [ x] .

) + 0 ≤ {x} < 1. . : {4,3} = 0,3; {2} = 0; {−12, 4} = 0, 6; {π} = π − 3 .

3. .

( " ' * " .

I.a,b a + b :

1)a + b = b + a ;

2)(a + b) + c = a + (b + c) ;

3)0 : a + 0 = a ;

4)(−a) : a + (−a) = 0 .

II.a,b a b = ab :

1)ab = ba ;

2)(ab)c = a(bc) ;

3)1: a 1 = a ;

4)a ≠ 0 1 : a 1 = 1 .

aa

III.a,b, c : (a + b)c = ab + ac .

IV. " *. a 2 *: a > 0 , a = 0 , a < 0 ; a > 0 , b > 0 : a + b > 0, ab > 0 ; b > a , " b + (−a) = b − a > 0 ; b = a , " b − a = 0 .

15

V. .*. " ! * " + + A , B -

, a A, b B a ≤ b , , u ,a A, b B : a ≤ u ≤ b .

4. ! .

. / + X , * " !, " M

, x X : x ≤ M .

. / + X , * " , " m

, x X : x ≥ m .

? M * , * " ! + X ,

m– + X .

., + ’, ! + , * " ! *

" ’, x *: x < 0 . # ! "

M + " *. / + ! + , *

" ! * " x *: x > 0 . "

m+ + " *.

6, " + X ! + , , -

* * (, * ! * " , " ! *2, + -M , + ! * ' ' + X . . " +

’, * ' ' ! * *, ( ! * "

. , " + X ! + , , -

* * + (. " + + * ' - ' ! ! * " ’, .

. .( 2 ( + X , * "

! + X , ! + X , * "

" sup X . .(! *2 + ( + X , * " -

' + X , ! ( + X , * " "

inf X .

5 M = sup X , :

1)x X : x ≤ M ;

2)ε > 0 x′ X : x′ > M − ε .

5 m = inf X , :

1)x X : x ≥ m ;

2)ε > 0 x′ X : x′ < m + ε .

+ + * + ( +, + * (

+. . , " X = [a,b) , sup X = b X . " X = [a,b] ,

sup X = b X .

16

6 " * ,, " + X , -

' *, " " * , (

(). , ( + " * +.

". ) & X ! ( ), -

! ( ) .

. .( + " + X ! + . %

" Y – + ( + X . ) * + X

' ! + , + Y + ". # x X , y Y

: x ≤ y . & ( . . 3) ,

β , x X , y Y : x ≤ β ≤ y . ) + β ! + , +

X , ! β , * ' ' + X . ) * y Y

β ≤ y , β , ( 2 ' * ' ' + X , !

β= sup X . * " " + * .

5.# . .

+ * " - +. , + , - "' * " ( +. ! " -

' + XIX . * (

. ) ( " " , -

+ +. 5 * , * "

(, + ' * " (! + *, - ' * ). ), + ! * -

+ ( * , ' * * * . " + " 2.

1. $" + A * +

B * . / + + , -

* ', * " :

1 → 2, 3 → 4, ... , 2n −1 → 2n,... .

# ! + , .

2. .( A – + * , B – +

* . & , * + :

1 → 2, 2 → 4, 3 → 6,..., n → 2n,... .

+ ( * , . +

+ B , + ' + A : B A . 0 ,' -

+ * !, + * + A , ! ! *2 - + + B . ’" , * ", + +.

3. .( AB CD , | AB | < | CD | . % +, + * + AB , +-

17

' + CD . # ! + ! * " -

, +.

& , !. $2 , AB CD *

. % " AC BD O -

( . 12).

	$ . 12.
. ( P	– * AB . % " OP
1	1

P2 , " CD . ), P2

, ,, + ( AB ( *

CD . # CD ! * Q2

" OQ2 . % Q1 , " -

AB . # Q1 + , ,, + + ( CD

( * AB . # + -

AB CD , *, +

+ +.

4. % +, + ! * " +

" +. .( AB " l . . -

AB " ( O – AB ) 2 -

" l 2 , " * , * AB

( . 13).

$ . 13.

18

. ( P	– * AB . ) P -
1	1

" " l P2 * "

. % " OP2 P3 ,

" " ' l . # , ,, + + ( AB

( * " l . .( Q3 – *

" l . % " OQ3 Q2

, " . ) Q2 " AB . ) , , Q1 . ) + + ( " l

( * AB , + AB " l , *, ,,

+ AB " l +.

/ + , + + * - + ", ( * , *

, . ) + * & «* +», *

* " .

4 ' *, * + -

"' * " ( +. 6 « »

+ " !.

" " +. %,

+ A B , +, ! + +

, *. 0 , , - + B1 + B , " + + A . # + *, -

B * A .

6. $ .

. / + , * " , " +

+ * .

# ! + + . % 1, 2

* ' *, + -

* – .

". ! – .

. & , * + +-

:

0 →1,1 → 2, −1 → 3, 2 → 4, − 2 → 5, 3 → 6, − 3 → 7,..., n → 2n, − n → 2n +1,... .

# * + . 6 + ' * .

1. + .

". .( M – * +. )! " (- ! * + M ( a1 . & + M 2 *-

" * * . )! ,

19

( a2 . 4 ( + + ! + – " + !-

" + M ( + ! 2 " -

* * . , +:

M1 = {a1, a2 ,..., an ,...} ,

" , + ' + M . & * 1 .

6 , ,, + , «( 2 '»

+. # !, " + , (

.

2. %’

.

. .( A – +. # + -

, ' + ":

A = {a1, a2 ,..., an ,...} .

. ( B – +:

B = {b1,b2 ,...,bp } .

6 2 + A B :

A B = {b1,b2 ,...,bp , a1, a2 ,..., an ,...} .

# , * + +

A B :

b1 → 1, b2 → 2,..., bp → p, a1 → p + 1, a2 → p + 2,..., an → p + n,... .

) + A B – +.

3. %’ ! ! .

. .( A B – +:

A = {a1, a2 ,..., an ,...} , B = {b1, b2 ,...,bn ,... } .

6 2 A B :

A B = {a1, b1, a2 , b2 ,..., an , bn ,... } .

# , * + +

A B :

a1 → 1, b1 → 2, a2 → 3, b2 → 4,..., an → 2n − 1, bn → 2n,... .

) + A B – +.

4. %’ ! -

.

( . .( A1, A2 ,..., Am

– +. 5 m = 2 , + " (* 3).

%, + " " m = k , ! +									%	=
									Ak
= A1 A2 L Ak		– . % +, + " "
m = k +1, ! + A1 A2 L Ak Ak +1 – . (:
A A L A A			= ( A A L A ) A			%	A	–
						= A
1 2	k	k +1	1 2	k	k +1	k	k +1

+ ( 3).

20

5. %’ !

.

. $" +:

A = {a		, a	, a	,..., a	,...},
1	11	12	13	1n

A2 = {a21, a22 , a23 ,..., a2n ,...},

A3 = {a31, a32 , a33 ,..., a3n ,...},

…

An = {an1, an2 , an3 ,..., ann ,...},

…

6 2 !’, " A1 A2 L An L :

A1 A2 L An L =

= {(a11), (a12 , a21), (a13 , a22 , a31),...(a1n , a2,n−1, a3,n−2 ,..., an1),...} .

+ ( , " +, ' * " , "

', , ! *2 1. 6 * , +

. (:

a11 → 1, a12 → 2, a21 → 3, a13 → 4, a22 → 5, a31 → 6,...

a	→	(n −1)n	+ 1, a	→	(n −1)n	+ 2,..., a	→	n(n + 1)	,...
1n		2	2,n−1		2	n1	2

% " , ':

21