{a n } M >0 s.t. a n 5 M for n =1, 2,... lim n a n = α ε =1 N s.t. a n α < 1 for n > N. n > N a n 5 a n α + α < 1+ α. M := max{ a 1,..., a N, 1+ α } a n 5 M ( n) 1 α α 1+ α t a 1 a N+1 a N+2 a 2 1
a n := ( 1) n ( n =1, 2,...). ( 1 n a n := n n 2
{a n } M >0 s.t. a n 5 M for n =1, 2,... {a n } a 1 5 a 2 5 5 a n 5 3
{a n } A {a n } A α := sup A α = sup a n n=1 (1) a n 5 α for n (2) ε > 0 α ε A N s.t. α ε < a N. {a n } n > N a n = a N α ε < a N 5 a n 5 α < α + ε a n α <ε for n > N 4
5
a 1 := n 1.4, a 2 := 1.41, a 3 := 1.414, a 4 := 1.4142, a n := max x = m o 10 n ; m N, x2 5 2 ( ) {a n } α a 2 n 5 2( n) α 2 5 2 b n := a n + 1 10n lim n b n = lim n a n = α a n ( ) b 2 n > 2( n) α 2 = lim n b2 n = 2 α 2 =2 α = a 1 > 0 α = 2 a n 6
r R ε > 0, q Q s.t. r q <ε r ε> 0 1 10n <ε n N r 10 n a n a n Q 0 <r a n < 1 10 n <ε. a n := µ 1+ 1 n n (n =1, 2,...) 7
a n = = = 5 < nx µ n 1 k n k=0 k nx n(n 1) (n k + 1) 1 k! n k=0 k nx µ 1 1 1 µ 1 k 1 ( ) k! n n k=0 nx µ 1 1 1 µ 1 k 1 k! n +1 n +1 k=0 n+1 X µ 1 1 1 µ 1 k 1 = a k! n +1 n +1 n+1 k=0 8
n P 1 ( ) a n 5 k=0 k! k = 2 k! = 1 2 k = 1 2 2 = 2 k 1 1 1 2 n 1 1 nx 1 a n 5 1+1+ 2 k=2 k 1 = 1 + 1 + 2 1 1 < 3. 2 {a n } µ lim 1+ 1 n = e n n 9
{a n } n +1 1, 1+ 1 n, 1+1 n,..., 1+1 n 1+n 1+ 1 n = n +2 n +1 n +1 = 1 + 1 n +1 r1 n+1 1+ n 1 n 1+ 1 r n +1 n+1 1 1+ 1 n n n +1 1+ 1 n+1 > 1+ 1 n n +1 n 10
a 1 >b 1 > 0, a n+1 := 1 2 (a n + b n ), b n+1 := a n b n a 1 >a 2 > >a n >b n > >b 2 >b 1 b n a n t b 1 b 2 b 3 a 3 a 2 a 1 (1) a 1 >b 1 a k >b k a k+1 b k+1 = 1 2 (a k + b k ) p a k b k = 1 2 ( a k p b k ) 2 > 0. n a n >b n (2) a n+1 = 1 2 (a n + b n ) < 1 2 (a n + a n )=a n, b n+1 = p a n b n > p b n b n = b n. 11
a 1 >b 1 > 0, a n+1 := 1 2 (a n + b n ), b n+1 := a n b n a 1 >a 2 > >a n >b n > >b 2 >b 1 {a n } {b n } α = lim n a n, β = lim n b n. n α = 1 2 (α + β) α = β lim n a n = lim n b n a 1,b 1 12
I n =[a n,b n ](n =1, 2,...) (1) I 1 I 2 I n (2) I n := b n a n 0(n ) T = 1 α s.t. I n = {α} n=1 (1) a 1 5 a 2 5 5 a n 5 b n 5 5 b 2 5 b 1 {a n } {b n } α = lim n a n, β = lim n b n (2) β α = lim n (b n a n )=0 β = α a n 5 α = β 5 b n ( n) α I n ( n) α T I n α b n t t t t t α 0 a n α α 00 n=1 13
(1) Dedekind (2) (3) (4) (1) Dedekind 14
{a n } b n := sup a p p=n b 1 := sup{a 1,a 2,a 3,...}, b 2 := sup{a 2,a 3,...},... b 1 = b 2 = = b n = {b n } lim sup a n lim n n a n c n := inf a p p=n c 1 5 c 2 5 5 c n 5 {c n } lim inf n a n lim a n n 15
λ := lim sup a n = lim n n sup a p = inf sup a p, p=n n=1 p=n b 1 = b 2 = = b n λ b n := sup a p p=n ε> 0 given (1) N s.t. n > N = λ 5 b n <λ + ε n > N a n 5 b n <λ + ε (2) p n = p s.t. λ ε 5 b p ε < a n p n λ ε < a n (1), (2) (3) λ {a n } {a nk } λ < λ 0 λ 0 t λ ε λ λ+ε λ 0 t 16
µ := lim inf n a n = lim n inf p=n a p = sup n=1 inf a p, p=n c n := inf p=n a p c 1 5 c 2 5 5 c n µ ε> 0 given (1) N s.t. n > N = µ ε < c n 5 µ n > N µ ε < c n 5 a n (2) p n = p s.t. a n <c p + ε 5 µ + ε p n a n <µ+ ε (1), (2) (3) µ {a n } {a nk } µ 0 <µ µ 0 µ 0 µ ε µ µ+ε t t 17
lim sup, lim inf (1) {a n } lim sup n a n :=. (2) {a n } lim inf n a n :=. ± lim sup n a n lim inf n a n lim sup n a n = a n = n (n =1, 2,...) lim inf n a n = 18
(1) a n := ( 1) n + 1 n lim sup a n =1, n 1+ ( 1)m m (2) a n = ( 1) m m lim sup a n =1, n lim inf n a n = 1. (n =2m) (n =2m 1) lim inf n a n =0. (3) a n := ( 1) n n lim sup n a n =, lim inf n a n =. 19
{a n }, {b n } (1) lim sup (a n + b n ) 5 lim sup a n + lim sup b n n n n (2) lim inf n (a n + b n ) = lim inf n a n + lim inf n b n (1) α := lim sup a n, β = lim sup b n n n ε > 0 given N s.t. n > N a n <α + ε, b n <β + ε. a n + b n <α + β +2ε ( n > N) n > N sup(a p + b p ) 5 α + β +2ε p=n lim sup n (a n + b n ) = inf sup n=1 p=n ε> 0 (1) (2) (a p + b p ) 5 α + β +2ε 20
(1) a n =( 1) n, b n =( 1) n+1 (2) {a n }, {b n } (3) ε a, b R given. ε > 0 a 5 b + ε a 5 b a > b ε = 1 2 (a b) a 5 b + ε a 5 1 2 (a + b) 21
{a n } {a n } lim sup n a n = lim inf n a n [= ] lim n a n = α ε > 0 given N s.t. α ε < a n <α + ε for ( n = N) n = N α ε 5 sup a p 5 α + ε, p=n α ε 5 inf p=n a p 5 α ε. lim sup n a n = lim inf n a n = α 22
{a n } {a n } lim sup n a n = lim inf n a n [ = ] lim sup a n = lim inf n n a n = α ε > 0 given. N s.t. a n <α + ε for n > N N N s.t. α ε < a n for n > N α ε < a n <α + ε lim n a n = α 23