Матан Лекции

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 2. #%! # & # 2 * +

[a, b) = {x Î —: a £ x < b},

(a, b] = {x Î —: a < x £ b}.

+* " ) ! " # " ) # a # b /+ / */ & #) ! #. .#+ ( 3 ) ! ( #/.

( # 1. & M # X Ì —, x Î X x £ M.

( # 1 + " ' & #%! - + + /( & )#* !" ++ (# !" / #' #) * & #0. % /" #( #/ # #$#%# + ' ) * ' * ! 3 ( + ( '7 #* 0 + (+ #3 3# !" + , ( 6#" #) * + " #* # 3 * # #* # * # (#: + +* 36 * # "x, 0 ) 6#' "( / +* " x", # + * 6 * + + #/ $x, 0 ) 6#' "* 6 * + x" # # "( / & x".

# 3 0, * & * ( # 1, M - + " // & #% 2 * + X, * #

"x Î X x £ M.

( # 2. & m # X Ì —,

"x Î X x ³ m.

( # 3. X Ì — , X #,

$M "x Î X x £ M.

X , X #,

$m "x Î X x ³ m.

30

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 2. #%! # & # 2 * +

X , X .

) +#( , & #) *+ " 2 * + # 3 * ) & + " #" & #%. '* +# , * # M - + " // & #% X # M ¢ > M, M ¢ - 2 + " // & #% X, "x Î X x £ M < M ¢. ) & #) * #0

2 * + # 3 * ) & #2 #" & #%. ( # * ! ) ! & #%! 2 * +.

( # 4. ' ( # X #,

X sup X.

#*. 1. " // & 2 * +.

( # 5. ' ( # X #,

X inf X.

* * 6 * + + #/ + " ' # #2 ' & ' 2 * + 0 * 2 + ( & ( & 3* 2( #/. 2(

(+ & ) * #0 ( #/ 4 # 5.

" // & a = sup X 2 * + X " #0 */ (+ / * +#/ #:

1."x Î X x £ a;

2."e > 0 $xe Î X xe > a - e.

. + * +# 0 ), ) a - + " // & #% 2 * + X. * +# 0 ), ) a 7 e,

7 )#* a - e * 3! + " ' & #% ' 2 * + X. ) " #0 #2 & .

b = inf X, * # +! / */ (+ * +#/:

1. "x Î X x ³ b;

31

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 3. * 6 * + + #/ + " ' & #

2. "e > 0 $xe Î X xe < b + e.
. + * +# 0 ), ) b - #2	// &	#% 2 * + X, + * +# 0 ), ) b + #) e,
3 7 )#* b + e * 3!	#2 ' & #% ' 2 * + X.
3. % &
%## 1 )#* ! 13 * + !" *+ '* + ( '* +# !" )#*. *# #) * ' ## 1 # *+ '* + * 2
( # 2 * + —, * # ( 3 +#		# * +# #+# * # 1 ¹ 0 # * ( 6 *# !.
14. */ * & #) *+ "		2 * + # + " & .

. * ! # / # * #+ % %# 2 * + — 2 * + +* " 3 * ) !" ( */ #) !" ( 3 ', 3/0 ! + # *+ '* + 14, # # + +#( *# !.

1. .

. 2 * + X *, 0 )#, # */ 1 x Î X. !3 % )#* m, 7, ) x.

2 3! + " ' & #% ' X, x > m. &, 2 * + X & #) *+ ", 1 # + " & #% M ¢. !3 % )#* M, 3 7, ) M ¢. #* M 2 /+ / */ + " ' & #% ' X.

# 3 0, +!3 ! (+ % !" )#* m # M, #0 !" m /+ / */ + " ' & #% ' X, M /+ / */ + " '

& #% ' X. 2( m # M " (# */ ) !'

3 % !" )#*. .3# / #" ) (#,

'( (+ * * ( #" % !"

)#* a0 # a0 + 1, #0 !" a0 /+ / */ + "

' & #% ' X, a0 + 1 /+ / */ + " ' &

#% ' X.

#*. 2. # 7 #0 % !" )#*, /+ / 6 */ + " ' & #% '.

0( # 0 [a0, a0 + 1] 10 + !" ) * '. .3# / ) # ( #/ ) (#, '( (+ * * ( # ) # a0 + 0.a1

# (a0 + 0.a1) + 1/10, #0 !" a0 + 0.a1 /+ / */ + " ' & #% ' X, (a0 + 0.a1) + 1/10 /+ / */ + " ' & #% ' X.

32

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, %#/ 3 3. * 6 * + + #/ + " ' & #

#*. 3. # 7 #0 )#* +#( a + 0.1b, &( a, b - % !, /+ / 6 */ + " ' & #% '.+ 0( # 0

[a0 + 0.a1, (a0 + 0.a1) + 1/10]

10 + !" ) * '. .3# / ) # ( #/ ) (#,

'( (+ * * ( # ) #

A0 + 0.a1a2

#

(a0 + 0.a1a2) + 1/100,

#0 !" a0 + 0.a1a2

/+ / */ + "

' &

#% ' X, (a0 + 0.a1a2) + 1/100 /+ / */ + " ' & #% ' X.

. ( 2 % **

& #)

. n- 7 & ( '* +#/ 0( # 0

[a

0

+ 0.a a ...a

n-1

, (a

0

+ 0.a

a

...a

n-1

) + 1/10n-1]

1 2

1

2

10 + !" ) * '. .3# / ) # ( #/ ) (#,

'( (+ * * ( # ) #

a

0

+ 0.a a ...a

n-1

a

n

#

(a

0

+ 0.a a

...a

n-1

a

) + 1/10n,

1 2

1 2

n

#0 !" + / /+ / */ + "

' &

#% ' X, + / /+ / */ + "

' & #% ' X.

. % ** ( 2 */ # (. # & ) */ ( '* +# )#* a, #0 3 2 3 * ) ' ( */ #) ' ( 3

						a = a0 + 0.a1a2...an...,
( / +*/ & & n ( + + / * +#/:
1. a0 + 0.a1a2...an					/+ / */ + " ' &	#% ' X;
2. (a	0	+ 0.a a	...a		) + 1/10n /+ / */ + "	' & #% ' X.
		1 2		n

33

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 3. * 6 * + + #/ + " ' & #

&, ( / 3 & & n * + ( #+! + * +

(3)	a		+ 0.a a ...a			£ a £ (a +0.a		a	...a	) +		1				,
											10n
		0		1 2	n	0	1	2	n
* + / ) * + (3) - 1 & # a				( *, + / ) * + (3) - 1 & # a #03!.

. 2, ) a = sup X. ) 3 (# */, ) a - + " // & #% X. '* +# , +*/ #' 1 x Î X + *" (#

3 ' + " ' & #%! X, 1 ( / 3 & n Î Õ +! / */ + * +

1

x £ (a0 + 0.a1a2...an) + 10n ,

+ *# + * + (3) + ) + * +

	x - a £ x - (a		+ 0.a a ...a		) £		1			.
						10n
		0	1 2	n
. # / * (	+ * + 1 # 0 ), )
	x - a £ 0,
* a - + " // & #% X.
2, ) a -	# 7 / #0 + " #" & #% X. '* +# , +*/ / + " // & #% b 2 * + X

+ *" (# )#*, /+ / */ + " ' & #% ' X, 1 ( / 3 & n Î Õ +! / */ + * +

b > a0 + 0.a1a2...an,

+ *# + * + (3) + ) + * +

a - b £ (a0 + 0.a1a2...an) +			1		-b <			1		.
			n					n
	10					10

. # / * ( + * + 1 # 0 ),)

34

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 4. 7 # 2( & / # 2 * +

a - b £ 0,

* a - # 7 / #0 + " #" & #% X. # 3 0, ! 0 #, ) a = sup X # ( 0 # * ! 1.

4. '

1 *$# + # ( 0 #7 ( / + " ' & # 2 * +. * 6 * + + #/ #2 ' & # 3

( ' $# + # # ) & ( 0 * +. * * #/ ' ! * +# * 7 #/ 2( & / # 2 * + # +!+ ( #2 ' & #, # /* 1.

/( * 2 * + X ** # 2 * +

-X = {x: (-x) Î X},

* * /6 #0 +* " 1 + 2 * + X, 2 !" (-1). 2 * + X # + " & #% M &( #

&(, &( 2 * + -X # #2 & #% (-M),		+ * + x < M # -x > -M + *# !. )
2 * + X #	#2 & #% m &( # &(, &( 2 * + -X # + " & #% (-m).
. ( 3 ! 3 0	3 (, ) 2 * + X # # 7 + " & #% a &( # &(, &(
2 * + -X #	#3 7 #2 & #% (-a), # X # #3 7 #2 & #% b &( # &(,
&( -X # # 7 + " & #% (-b).
. + ( ! ** 2( #/ 3 2( + * + ( #+ * # * 7 #'
	sup X = - inf (-X),	inf X = - sup (-X).

2. .

. .* 2 * + X * # & #) * #0. &( 2 * + -X 2 *, & #) *+ ". . 1 2 * + -X # + " & a, * ( + , 2 * + -X # #2 & -a.

2 ( 0 .

35

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 3 4. 7 # 2( & / # 2 * +

!& /(/ + 0 ! # * ( 6# ( & + * # + ( ## + " #" # #2 #" & ' & #) & # # * & 2 * +.

1. .*	2 * + X	&	#)	*+ ". &( & sup X = ¥.
2. .*	2 * + X	&	#)	* #0. &( & inf X = -¥.

3. .* X = Æ. &( & sup X = -¥ # inf X = ¥.

36

com.neevia.http://www version trial Converter Personal Neevia by Created

			, %#/ 4
		1.	.( * ( + * #
		4
1.	.( * ( + * #..................................................................................................................................................		37
2.	(# * +	* ( *" (/6 '*/ * ( + * #...............................................................................................	38
3.	& #)	* *" (/6 '*/ * ( + * #.............................................................................................................	39
4.	" (# * ' & #) ' * ( + * # ..........................................................................................		40

1. ( "

( / #* #) * " ( 0+# #/ * + #' #) * & #0, # # #+ & # #* & + * #/ #// #/ (, # / & 1 3 ) #/, '( ) * # * & ( #, + *" (/6 7#.

( # 1. & l x1, x2, ..., xn, ...,

"e > 0 $n0 "n > n0 |xn - l| < e,

l = limn®¥xn.!, , , -

.

#* e > 0 * 2# ' 3 #0 * # xn # l. ( # 1 * + #+, ) xn #3 #2 */ l * 3 ' ' ) * #, )# / * & n0. * * + , n0 0 +#*# e.

3 # + # #, ) + +* 36 * # "e > 0 ( * * ! 3 36 #/ ( #/ 1. , * C > 0 -

$# *# +	)#*. * # e # # +* 2# ! 0 ) #/, Ce 2 # # +* + 0 2 !
2#	! 0 ) #/. .1 ( # 1 2 $0# + + * ( 6 +#(: limn®¥ = l, * #
	"e > 0 $n0 "n > n0 |xn - l| < Ce.

!& ( * & & ( #/ 0 ) */, # ) &, + # + ## # # +# +!+ ( + #) * #

( 0 * +. / 2 #/ + ) * ' 1 +# #3 #* # ( (# #% # ( #/ 1: l

x1, x2, ..., xn, ...,

37

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 4 2. (# * + * ( *" (/6 '*/ * ( + * #

$e > 0 "n0$n > n0 |xn - l| ³ e.

. 2 1 # # ( #/ 1 " 2( # ( + * !" * ( + * '. .*, #, xn = 1/n # '(, )

l = lim

1

= 0.

n®¥ n

+ * + |xn - l| < e * +# */ 1 +#+ ! + * +

1

< e # # n >

1

.

n

e

!3# / + ) * + n0 % )#*, 3 7, ) 1/e, 3 2( */ + +! ## ( #/ 1.

) ! 3 0 " (#, + ) * * #, )

lim

1

= 0, (k > 0),

lim

1

= 0,

lim

1

= 0, (a > 1).

n®¥ nk

n®¥ 10n

n®¥ an

2. ) " & "

) * ( 0 * + (# * + * # ( *" (/6 '*/ * ( +

* #.

1. ) , .

. .* *" (/6 /*/ * ( + * x1, x2, ..., xn, ... # ( ! l # p. & * ( # 1 # # l

(1)	"e > 0	$n1 "n > n1	|xn - l| < e.
) # # ( # 1 ( p
(2)	"e > 0	$n2 "n > n2	|xn - p| < e.

38

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 4 3. & #) * *" (/6 '*/ * ( + * #

#* n1 # n2 + (1) # (2) + 36 * ) 0 !. : / ( + & +! #/ (1) # (2), 3 0 )# n0 = max{n1, n2} # 3 ( ** #+ n > n0. / #" n +! ! (1) # (2) # * + ( #+! + * +

|l - p| = |l - xn + xn - p| £ |l - xn| + |xn - p| < e + e = 2e

# #, )# !+ / #7 ' # 0+ / % ) #, 0 ), ) ( / 3 & e > 0 * + ( #+ + * +

|l - p| < 2e.

) * * #, +!3# / 3! n Î Õ # e = (1/2)(1/(10n)), 1 %## 3 +#(#, ) |l - p| £ 0. * ( )#*

2 3! #% !, |l - p| = 0 # 0 )#, l = p, ) ( 0!+ 1.

3. " & "

. ( 2# * # ## * ( + * ' #** ( + # +0 # *+/0 ' 2( 0 #) ! # *+ '* + # # #,

*" (# *, & #) *, * # ( #. ) 2, ) *" (# * * ( + * # + )

&	#)	*, # # ( # * + #, 3 + # *" (# * # /+ / */ 3 2 * # 7 # 3 + #
&	#)	* # * ( + * #.

2. .

. .* * ( + * x1, x2, ..., xn, ... *" (# */. 3 0 )# ( ) 0 l # 0 #7 ( # 1

"e > 0 $n0 "n > n0 |xn - l| < e.

- $# *# e > 0. + * + |xn - l| < e + *# *#* (+ " + * + -e < xn - l < e # # l - e < xn < l + e, / 0 ), ) +* xn * # n > n0 ( + # + (l - e, l + e).

#*. 1. * %#/ 3 & #) * # *" (/6 '*/ * ( + * #.

# 3 0, 2 * + 1 +

39