: Q R.
: > > ³ ³
³ / 2 + / 2 > / 2 + / 2 > / 2 + / 2 >
: > / 2 + / 2 > , ( / 2 + / 2)Q
.
,bÎR <b. 1-5 - :
1) M ÎR: ££b (<<b) - ()
2) M ÎR: £<b (<£b) - ()
3) M ÎR: < & x<b -
4) M ÎR: £ & £b -
5) M ÎR -
: b ( ) ( ), b-a
: aN , bN , "n aN£bN (bN-aN)-, $! : "n cÎ[aN,bN] ( Ç[aN,bN])
:
aN£bN£b1 aN & aN£b1 => $ Lim aN=a
a1£aN£bN bN & a1£bN => $ Lim bN=b
aN£a b£bN aN£bN => a£b
Lim (bN-aN)=b-a=0( )=>a=b
c=a=b, aN£c£bN
: aN£c£bN, ¹
aN£c£bN=>-bN£-c£-aN => aN-bN£c-c£bN-aN => ( ) => Lim(aN-bN)£Lim(c-c)£Lim(bN-aN) => (a-b)£Lim(c`-c)£(b-a) =>
0£lim(c`-c)£0 => 0£(c`-c)£0 => c=c => c - .
.
, .
{n},{yn}
{xn+yn}
ε>0 N1 n>N1 |xn|<ε/2
N2 n>N2 |yn|<ε/2
N=max{N1,N2} n>N
|xn|<ε/2 |yn|<ε/2
|xn+yn|<=|xn|+|yn|<ε/2+ε/2=ε
, .
{n},{yn}
{xn+yn}
>0 N1 n>N1 |xn|</2
N2 n>N2 |yn|</2
N=max{N1,N2} n>N
|xn|</2 |yn|</2
|xn+yn|<=|xn|+|yn|</2+/2=
.
.
{n}
{yn}
>0 |yn|<M
ε>0 N n>N |xn|<ε/M
|xnyn|=|xn||yn|<ε/M*M=ε
.
{xn} n xn≠0 {1/xn}
N n>N |xn|>1 {1/xn}
Ε>0 N n>N |xn|>1/ε <=>|1/xn|<ε
{xn} xn≠0 n, {1/xn}
20. . .
{xn} {yn}: |yn|<=|xn| => {yn}
|
|
|
Ε>0 N n>N |xn|<ε
|yn|<=|xn|<ε
{1/n^λ} λ>0
ε>0 N n>N
N=(1/ε)^1/λ => n>(1/ε)^1/λ <=> n^λ>1/ε <=> 1/n^λ<ε
{xn} , {xn-a}
, xn->a, lim xn=a
xn->a, yn->b
xn+yn->a+b
(xn+yn)-(a+b)=(xn-a)+(yn-b)
, .
=
xn->a, yn->b
xn*yn->a*b
xn*yn-a*b=xn(yn-b)+b(xn-a)
=
,
xn->a, xn->b
{xn-a},{xn-b}
{(xn-a)-(xn-b)}={b-a} b-a=0
Vε>0 N n>N |xn-a|<ε
.
1 {xn} xn>=b
lim xn>=b
lim xn=a<b
ε=b-a
|xn-a|<ε=b-a
a-b<xn-a<b-a
xn<b
xn<b? => lim xn>b? lim=0
2 xn, yn
xn<=yn
limxn<=limyn
yn-xn>=0
lim(yn-xn)= limyn-limxn>=0
limyn>=limxn
3 {xn},{yn} lim
xn<=zn<=yn zn limzn=limxn=limyn
limxn=limyn=a
xn-a<=zn-a<=yn-a zn-a lim zn=0
.
x1, x2, xn
, .
