шамин с сдо

{an}

an =

n2

.

n2 + 1

lim an = 1

n→∞

ε

N = N(ε)

n > N

2

− 1 < ε.

n2n+ 1

n2

1

=

n2

− n2 − 1

=

1

.

n2

+ 1

−

n2 + 1

n2

+ 1

1

< ε.

n2 + 1

n2 > ε−1 − 1.

N(ε)

√ε−1 − 1, ε < 1.

N(ε) =

1,

ε ≥ 1,

A

A ≥ 0

n

A ≥ 0

ε < 1

A < 0

|1 − A| = |A| + 1 > 1,

A < 0

ε < 1

a

b

b

a

̺(a, b) ≤ ̺(a, c) + ̺(c, b)

a b

c

x0

ε > 0

ε

x0

ε

x0

{an}

ε

an

a

b

a < b.

ε = b−a

ε > 0

2

ε

U(a, ε) U(b, ε)

N = N(ε)

n > N(ε)

an U(a, ε)

lim an = lim cn = d,

n→∞

n→∞

bn

lim bn = d

n→∞

ε > 0

N = N(ε)

n > N(ε)

bn =

sin n2

.

n

1

1

an = −

, cn =

.

n

n

lim an =

lim cn = 0.

n→∞

n→∞

|bn| =

sin n2

≤

1

,

n

n

an ≤ bn ≤ cn.

lim

sin n2

= 0.

n

n→∞

n

M

{an}

a

ε

ε = 1

N

|an − a| < 1,