матан Бесов - весь 2012

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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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a% ε a = +∞ R a = −∞ R

a = ∞ R " Uε(+∞) =

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X Y

X ↔ Y

◦ x X

y Y x → y

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◦ y Y x X x → y

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% ! ! "

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				−1		−2		−3
	m
n		0
		01	1	−1/1	1	−2/1	1	−3/1
		01	1	−1/1	1	−2/1	1	−3/1
		01	1	−1/2	1	−2/2	1	−3/2
		01	1	−1/3	1	−2/3	1	−3/3

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R R

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an A %

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! # .		lim an = a an → a # n → ∞
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) # lim	1	= 0

n→∞ n

+∞ −∞ ∞
			ˆ
R = R {−∞} {+∞}
			R = R {∞}

! ε > 0 ε a

" Uε(a) = (a − ε, a + ε) # $

x > 1ε Uε(−∞) = x & x R x < − 1ε Uε(∞) = x & x R |x| > 1ε

' U (a) a ˆ ε

() ! !

a ˆ

R "

{an} ε > 0 nε N& an Uε(a) n nε

+ ) !,

a ˆ

R "

{an} , U (a)

* * , "

" R - !

* *

, {n} .

R R R

ˆ R

/ * {n} *

R * R

/ * {(−1)nn} *

ˆ ∞

0" ) ! "* *

! a

{an} 1 1

' a {an}

ε0 > 0& n0 N n N n n0& |an − a| ε0

-2 ) !

. *

{an} "

!* "* a, a R

! ε > 0 Uε(a) ∩ Uε(a ) = 3

, nε N nε N

an Uε(a) n nε an Uε(a ) n nε

n¯ε = max{nε, nε} ! an Uε(a) ∩ ∩ Uε(a ) n n¯ε

! 3

{an} "

*! !

{an} "

b R : |an| b (an b, an b) n N.

= ab

{an}

a = lim an ε = 1 n1 N |a − an| < n→∞

< 1 n n1

a − 1 < an < a + 1 n n1.

b1 max{a + 1, a1, a2, . . . , an1−1}

{an} ! b1 " #

{an} {an}

! $% %

& '( ) ' ! *

! ! ! + a, b R

%◦	an bn cn n N lim an =		lim cn = a
	lim bn = a,	n→∞	n→∞
	lim bn = a,
	n→∞		n nb,
$◦	lim an = a a < b nb N an < b		n nb,
-◦	n→∞	b n N a b lim an = a
-◦	lim an = a an	b n N a b lim an = a
	n→∞		n→∞
	an b n N a b

lim an = a |an| b n N |a| b

n→∞

) 3◦ #

! <

lim an = a
R	lim bn = b R	n→∞

n→∞		lim (an − bn) = a − b
%◦	lim (an + bn) = a + b
	n→∞	n→∞

$◦

lim anbn = ab

-◦

n→∞

= 0 n N b = 0

lim

n→∞ bn

! . ) 3◦

* ! αn = a − an βn = b − bn

αn

→ 0 βn

→ 0

n → ∞ ) 1◦ / ! ! *

! ! !

abn − ban

n =

−

bbn

|a(b − βn) − b(a − αn)|

|a|

|βn| +

|αn|.

|bbn|

|bn|

|bbn|

+ ! ! ε > 0 nε, nε , nε

N |αn| < ε n nε

|βn| < ε

n nε

|bn| = |b − βn| >

|b|

n nε

* ! nε = max{nε, nε , nε }

2|a|

ε +

|b|

ε = M ε

n nε

n ! M ε

0 . n N #

' lim an n→∞ bn

{αn}

lim αn = 0 n→∞

& ! 1! ) #

)

! " !

! 0 ! #

) {αn} {βn} βn = 0 n N
lim	αn	= 0	lim	αn	= ∞	lim	αn	= 1	lim	αn

n→∞ βn			n→∞ βn			n→∞ βn			n→∞ βn
(

lim an = a

n→∞

an = a + αn n N {αn}

{an}

lim an = ∞

n→∞

! {an} = {n + (−1)n} {bn} = {n}

{an − bn} = {(−1)n}

" # ˆ

$ %

" $# % #

& ' ' ()

sup{an} inf{an}

* # R " )) # ))

{an}

$ % an an+1 $an an+1%

n N + an ↑ $an ↓%

{an} ), $( ), %

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)

{an}

$ % an < an+1 $an > an+1%

n N

+ ), ( ),

)

. / -),

) # ( ),

§ e

{an} R lim an = sup{an}

n→∞

{an}

+∞

{an}

( a	sup{an} +∞
0 ) "	an a n N
ε > 0 nε N! anε Uε(a) ( anε an a
n nε (
an Uε(a) n nε.

1 lim an = a

n→∞

+ (

() ( ( ),

( {[an, bn]} ),

# " ξ $% , "

0 {an} ), {bn} ( ),

( lim an = ξ , )

n→∞

) sup{an} = ξ

( inf{bn} = ξ

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e lim				1	n
		1 +		1
		1 +	n
	n→∞		n
# ' (, ( 2(
!
(1 + h)n > 1 + nh		h > 0,				n N, n 2. $/%
	$/% (

( 3

4 () {xn}
		n+1	n + 1
1			n + 1
xn = 1 +	n	> 1 +	n	> 2 *

{xn}

1 +

n−1

n − 1

n+1

n + 1 (n + 1)(n − 1)

1 +

n2 − 1

n + 1

xn−1

n3 + n2 − n

1 +

n2 − 1

n3 + n2 − n − 1

> 1.

n + 1

lim xn [2, x1] = [2, 4].

n→∞

	xn			lim xn
				n→∞
e lim			=				= lim xn,
		1
n→∞ 1 +				lim	1 +	1	n→∞
		n
						n
				n→∞

" #

$ e % &

e= 2,718 . . .

' {nk} % !

# (

{ank } = {ank }∞k=1

{an} = {an}∞n=1#

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* ) , + # # #

$ #

" -

! R#

. μ R " U (μ)

/ " / .

0 )#

1 / 0

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U (μ) / .

# 1 μ

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" . xn1 U1(μ)

" . xn2 U 1 (μ)

! n2 > n1# 3 / U	1	(μ)
2

/ " / . # 4 "

xn3	U 1		(μ) n3 > n2# ' / &
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{xnk } ! R μ

" ε > 0 Uε(μ) /

. kεkε > 1ε #

' 0# #

5 6 #

{rn}

[0, 1]


{an} R a {an	k	}

U (a)

! {an}"

" !

{ank } # " lim ank = a

n→∞

{an}

$ $ % a

" lim an = a "

n→∞

a & % '

ε0 > 0 " Uε0 (a)

% !

{ank }" ! $

Uε0 (a) ( ) ' '

* +,$ -." $

{ank } $ " '$

{an}

a" ank Uε0 (a) k N"

{an} # " a = lim an

n→∞

) .

{an} - $ )- $.

/% lim an ) lim an.

n→∞ n→∞

, $ $

" 0 1 0

xn 0" yn 0 n N"

{xn} ) $ ."

{yn} $ $

" %


	lim xnyn =	lim xn lim yn.
n→∞		n→∞ n→∞

% 2 3

& * +,$ -

'$ $

! R"

) % .

{an} X = {x3 x R"

x % !

}

! X = 4 "

U (−∞) ! "

lim an = −∞ # " −∞ $

n→∞

$ {an}" a =

= lim an

n→∞

! X = & % sup X b" −∞ < b +∞

" b = lim an , ε > 0"

n→∞

bε Uε(b)" bε < b & % $ % " $ xε X3 bε < xε b

bε

{an} b > b b X

Uε(b)

{an}

ε > 0 b

{an}

! " b #

{an} b = lim an $"

n→∞

b > b & " " U (b )

b < b < b b (

# ) * "

b = lim an

n→∞

$ + ,-. # % / " /% " *

0 .

1 /% *

% / " %

{an} "

" ! 2

ε > 0 nε N : |an − am| < ε n, m nε. (3)

{an} * " lim an = a . !
n→∞
ε > 0 &
nε N : \|a − an\| <		ε	n nε.
nε N : \|a − an\| <	2		n nε.
	2
n, m nε
\|an − am\| \|an − a\| + \|am	− a\| <			ε	+	ε	= ε,
\|an − am\| \|an − a\| + \|am	− a\| <				+	2	= ε,
			2			2

{an} 4

" / (3)

* "

3 # {an}

. ! ε = 1 & (3)

n1 N : |an − an1 | < 1 n n1,


	\|an\| < 1 + \|an1 \| n n1.
{an}
5 #	+ ,-. # {an}
	+ ,-. # {an}
* "% / " {an		k	}

a lim ank
k→∞
6 # a " " "
{an} ε > 0 & (3)
nε, kε N : \|an − ank \| <	ε	n nε, k kε.
	2

* " k → ∞


\|a	− a\|	ε	< ε n n		.
\|a	− a\|		< ε n n		.
n		2		ε
		2

. ε > 0 lim an =

= a

n→∞

In = [an, an) = an, an + 10n

an 0 an = α0, α1α2. . .αn n α0 N0

αi {0, 1, . . . , 9} i 1

{In} = {In}∞n=0 = {[an, an)}

an ↑ an ↓ an − an = 101n → 0 n → ∞

a 0 ! n0 N" n0 a #$% α0 N0" α0 a < α0 + 1 & %

I0 [α0, α0 + 1)

I1 '$ a"

I1 = α0,α1; α0,α1 +		1	a.
I1 = α0,α1; α0,α1 +	10		a.
	10

& % I1 ()

I2 '$ a"

	1
I2 = α0,α1α2; α0,α1α2 +	1	a.

	100

{In} In =

= [an,

) an = α0, α1. . .αn

= an +

In a n

N * an an n

a → {In} = {[an, an)}. (

,-' (

{a R : a 0} ←→ Ω. .

α0, α1α2. . . α0 N0 αi

{0, 1, 2, . . . , 9} i N

& /0 "

		{In} → α0, α1α2 . . . ,					{In} Ω,	1

1						an = α0, α1
In =	an	,	an	+	10n		α2. . .αn	a 0

2 ( 1 ' $

a → α0, α1α2 . . . (a 0) 3

a → {In} → α0, α1α2 . . .

4 * '$ '$ $

a '

/0 $ '$

$ 5 1

{an} '

n0 N : an0 = an0+1 = an0+2 = . . .

{In} Ω

{an}

! !

6 ' a In n N 7-

a <	an	n N, lim	an	= a,
		n→∞

{an}

{¯ }

In =

{ } ¯

= [an, an] a In

n N a an n N ! an0 = a

n0 {an} " #

a < an n N a In n N

% & ' '

( ) *

+ ,

Ω ←→ { } +-,

. {In} Ω

{an} .

'/ {In}

+ , 0

n0 N n n0 In+1

In % 1 {an}

' 2

+ ,

Ω→ { }.

{In} {In} '

{In}

α0, α1α2. . . "

{In}

= {[an, an)} an = α0, α1α2. . .αn an = an +

10n

n N

{an}

1 n0 *

4 #

{In} Ω 5 + ,

{In}

+6,


	{a : a R, a 0} ↔
	{a : a R, a 0} ↔

. +7, +-,

+6, R

1 (

−a < 0 +a > 0,

−a → −α0, α1α2 . . . ,

a ↔ α0, α1α2 . . . +6,

(−a)n = −α0, α1. . .αn −			1n		(−a)	n =
			10

= −α0, α1. . .αn '

n −a

5 &

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