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

{xn0 +1 , xn0 +2 , ...,xn ,...}

& #) , * * ( 2# */ + # + (l - e, l + e). 2 * + +* " 1 + * ( + * # /+ / */ 35 (# # ) * '

( # 2. ! x1, x2, ..., xn,... · , "n Î Õ xn £ xn+1; · ,

, "n Î Õ xn < xn+1; · , "n Î Õ xn ³ xn+1; · (, ,

"n Î Õ xn > xn+1; · , ; · ,

.

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

3. .

. # + ! 3 * ( 2# (+ ( 2 #/: ( ( / 3!+ 6#" * ( + * ', ( & - ( / + 0 * 6#". # #) * 3 #" ( 0 * + 3 ** , 1 #03, #, * ) ' 3!+ 6 ' * ( + * # x1, x2, ..., xn..., & /, ) + ' * ) ' ( 3 .

* * ( + * x1, x2, ..., xn, ... & #) , 2 * + 1 + # + " & . 3 0 )#

a = sup xn

n³1

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

# ( 2, )

a = lim xn .

n®¥

#* a ( + + / (+ * +#/:

1."n xn £ a;

2."e > 0 $xn0 xn0 > a - e.

*# * # * ( + * # 0 ), ) "n > n0 xn ³ xn0 .

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

"e > 0 $n0 "n > n0 a - e < xn £ a.

#* + * + a - e < xn £ a + ) + * + |xn - a| < e, 1 * ( ( 2 # 0 ), ) a = limn®¥xn, ( 0!+ 6 3.

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

lim xn = inf xn .

n®¥ n³1

( 36 0 ) # , ) +* ( ! 0 * #, 3 ( ! * ( + * #, /+ / */

( # 3. * , limn®¥xn = +¥,

"M $nM "n > nM xn > M.

# #) ( # (, + !' -¥.

( # 4. * , limn®¥xn = -¥, * #

"m $nm "n > nm xn < m.

. # # / + + # # ( #/ 3 # 4 # + ( ! ** 2( #/, 0 ), ) * # * ( + * x1, x2, ..., xn,

... 3!+ # & #) , limn®¥xn = +¥;

42

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

* # * ( + * x1, x2, ..., xn,... + 0 * # & #) , limn®¥xn = -¥.

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( # 2. ! x1, x2, ..., xn, ... , limn®¥xn = 0.

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

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

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, %#/ 5 2. 4 * ) ! * ( + * # # #" *+ '* +

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

+ #+ / ( #/ * ( + * #, # 6 ' ( l, # 3 * ) ' * ( + * #, #" (#

0 ) #:

x1, x2, ..., xn, ... l , x1 - l, x2 - l, ..., xn - l, ... .

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

1. +

.

"e > 0 $n0 "n > n0 |xn + yn| £ |xn| + |yn| < e + e = 2e,

!' 0 ), ) limn®¥(xn + yn) = 0 # ( 0!+ + + 2( # ! 1.

2. .* * ( + * x1, x2, ..., xn, ... - 3 * ) /, * ( + * y1, y2, ..., yn, ... - & #) /. 8 0 ), )

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!' 0 ), ) limn®¥(xnyn) = 0 # ( 0!+ + + 2( # ! 1.

0 + & ) * # ! 1 # ( #+ ! # ** 2( #/ # * (* + +!+ (#, ) * ) & )#* 3 * ) !" * ( + * ' * */ 3 * ) '.

3 # + # #, ) + + ' ) * # ! 1 + 2( */, ) #0+ ( # 3 * ) !"

* ( + * ' /+ / */ 3 * ) !. ( '* +# * # + + 2( # ! 1 - 3 *# .

1. ! .

. 4 * ) ! * ( + * # /+ / */ *" (/6# #*/ # 1 & #) ! #. - )#,

#0+ ( # (+ " 3 * ) !" * ( + * ' 2 ** #+ #0+ ( # (+ " * ( + * ', #0 !" ( 3 * ) /, ( & / & #) /. &( #0 ! 1 * (, ) 1

#0+ ( # /+ / */ 3 * ) !. (* +# 1 ( 0 .

3. * + & "

6 ! 1 * #$#) * # ( '* +#/ ( * ( + * / # ) ! ( '* +#/ ( #"

( #.

1. ! limn®¥xn = l limn®¥yn = p. "

lim(xn + yn ) = l + p;

n®¥

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%#/, 5 3. #*#)#$ +#/'*( ( #*/(/6#*" +(* #/*

lim(xn - yn ) = l - p;

n®¥

lim(xn yn ) = lp.

n®¥

) , yn ¹ 0 n p ¹ 0,

( + ) * + ! # #) * + !. -#, ) * +#/ ! 1 1 +#+ ! , ) * ( + * #

x1 - l, x2 - l, ..., xn - l, ... # y1 - p, y2 - p, ..., yn - p, ...

3 * ) !.

0( # ( 0 * + 2( #$#) * ( '* +#.

1. . 1 0 ), ) *

x1 + y1 - (l + p), x2 + y2 - (l + p), ..., xn + yn - (l + p), ...

(+ " 3 * ) !" * ( + * ' /+ / */ 3 * ) ', 0 )#, limn®¥(xn + yn) = l + p, ) ( 0!+

+ + 2( # ! 1.

2. 2# 3 * ) * ( + *

y1 - p, y2 - p, ..., yn - p, ...

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%#/, 5 3. #*#)#$ +#/'*( ( #*/(/6#*" +(* #/*

& #) * ( + * -1, -1, ..., -1, ... # )# 3 * ) * ( + *. . 1 0 ), ) *

x1 - y1 - (l - p), x2 - y2 - (l - p), ..., xn - yn - (l - p), ...

(+ " 3 * ) !" * ( + * ' /+ / */ 3 * ) ', 0 )#, limn®¥(xn - yn) = l - p, ) ( 0!+ + + 2( # ! 1.

3. .0 /* $'

xnyn - lp = (xn - l + l)(yn - p + p) - lp = (xn - l)(yn - p) + p(xn - l) + l(yn - p),

+ * " * ( + * '

(x1 - l)(y1 - p), (x2 - l)(y2 - p), ..., (xn - l)(yn - p), ...,

p(x1 - l), p(x2 - l), ..., p(xn - l), ... # l(y1 - p), l(y2 - p), ..., l(yn - p), ...,

#0 !" + / /+ / */ #0+ ( # (+ " 3 * ) !" * ( + * ', + / # / - #0+ ( #/ * / !" ( * & #) !") # 3 * ) !" * ( + * '. ( + , +* # * ( + * # - 3 * ) !. . 1 * ( + * (5) - 3 * ) /, limn®¥(xnyn) = lp, ) ( 0!+

+ 2( # ! 1.

4. .0 /* $'

xn - l = pxn - lyn = p(xn - l) - l( yn - p) , yn p pyn pyn

0 ), ) * ( + *

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%#/, 5 3. #*#)#$ +#/'*( ( #*/(/6#*" +(* #/*

. * limn®¥yn = p,

"e > 0 $n0 "n > n0 |yn - p| < e.

0 + * +

|p| - |yn| £ |yn - p| < e

* (, )

|yn| > |p| - e.

#, ) p ¹ 0. !3

e = |2p| > 0

# " (#, ) ( / n > n0 * + ( #+ + * +

|yn| > |p| - |p2| = |p2| ,

* ( + ,

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