( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x
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1 Z Z Ẑ (X, d) X x 1, x 2,, x n, x x n x(n) ( ) X x x ε N N i, j i, j d(x(i), x(j)) < ε ( ) X x x n N N i i d(x(n), x(i)) < 1 n ( ) X x lim n x(n) X x X () X x, y lim n d(x(n), y(n)) = 0 x y x y 1
2 ( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x(2), y(2)),, d(x(n), y(n)), R lim d(x(n), y(n)) n (1) d ˆ : ˆX ˆX R d([x], ˆ [y]) = lim d(x(n), y(n)) n well-defined (2) ( ˆX, ˆ d) (X.d) (3) X x [x] = lim x(n) n 2
3 ( ) X x n m d(x(n), x(m)) < 1 n x ( ) X x ( ˆX ) ˆX [x 1 ], [x 2 ],, [x n ], ˆX lim n [x n] ˆX x n [x 1 ], [x 2 ],, [x n ], [x 1 ], [x 2 ],, [x n ], X y y(n) = x n (n) (n = 1, 2,, ) y ε n m n ε > 3 n [x 1 ], [x 2 ],, [x n ], ˆ d([x n ], [x m ]) < 1 n k n < m < k d(x n (k), x m (k)) x n, x m < 1 n d(x n (n), x n (k)) < 1 n, d(x m(m), x m (k)) < 1 n d(y(n), y(m)) = d(x n (n), x m (m)) d(x n (n), x n (k)) + d(x n (k), x m (k)) + d(x m (k), x m (m)) < 1 n + 1 n + 1 n < ε y ε, n.m, k d(y(k), x m (k)) = d(x k (k), x m (k)) 3
4 d(x k (k), x m (k)) + d(x m (k), x m (m)) < 1 n + 1 n m < k k ˆ d(y, x m ) 2 n < ε ε > 0 n s.t. m s.t. n < m ˆ d(y, x m ) < ε y = lim n x n 1.2 M {N n } N n M N 1 N 2 N n N n N n = {0 M } N 0 = M M {N n } N n M () d : M M R 0 x = y d(x, y) = 1 N n 1 x y N n n y n M x, y N n d(x, y) < 1 n N n x y M (M ) (M, d) M x, y, z (0) d(x, y) 0 (1) d(x, y) = 0 x = y (2) d(x, y) = d(y, x) (3) d(x, z) max{d(x, y), d(y, z)} d(x, y) < d(y, z) d(x, z) = d(y, z) (0),(1),(2) (3) n d(x, y) 1 n d(y, z) 1 n 4
5 N n 1 x y, y z N n 1 N n 1 x z d(x, z) 1 n d(x, z) max{d(x, y), d(y, z)} d(x, y) < d(y, z) d(x, z) max{d(x, y), d(y, z)} = d(y, z) d(x, z) < d(y, z) d(y, z) max{d(x, y), d(x, z)} < d(y, z) d(x, z) = d(y, z) M () (0) n N a a 1 n a + N n (1) M M 1.3 ˆM ˆM M a b a + b (a + b)(i) = a(i) + b(i) i = 1, 2, 3,, n, (1) M (2) M 0 M 0 M 0 0(i) = 0 M i = 1, 2, 3,, n, M (3) (1) (2) ˆM 5
6 ˆM (1) a b M a + b M (2) 0 M M (3) a M a ( a)(i) = a(i) i = 1, 2, 3,, n, a M a M a (4) a b M 0 M a b 0 M (5) a b M a b a b 0 a M d([a], ˆ [0]) = 0 [a] = [0] s = d([a], ˆ [0]) > 0 s = lim n d(a(n), 0 M ) 0 < s 1 n 1 n + 1 < s 1 n < 1 n 0.5 ˆ d([a], [0]) s = lim n d(a(n), 0 M ) N N s.t. N m s.t. N m 1 n + 1 < d(a(m), 0 M ) < 1 n 0.5 m d(a(m), 0 M ) = 1 n ˆM [a] [a] [0] N n, N s.t. N m s.t. N m d(a(m), 0 M ) = 1 n d([a], ˆ [0]) = 1 n 6
7 a N n, m n m d(a(n), a(m)) < 1 n a a M [a] [0] d([a], ˆ [0]) = 1 n d(a(n), 0 M ) = 1 n ( =) m n < m a d(a(n), a(m)) < 1 n d(a(n), 0 M ) = 1 n d(a(m), 0 M ) = 1 n (= ) N n, N s.t. N m s.t. N m d(a(m), 0 M ) = 1 n m N n N < m d(a(m), 0 M ) = 1 n a n < m d(a(n), a(m)) < 1 n d(a(n), 0 M ) = 1 n ˆM [a], [b] ˆ d([a], [b]) = ˆ d([a] [b], [0]) ˆM [a], [b] 0 [a] = [b] d([a], ˆ 1 [b]) = N N s.t. N m s.t. N m n d(a(m), b(m)) = 1 n a b a b a b d([a], ˆ 0 [a] = [b] [b]) = 1 d(a(n), b(n)) = 1 n n ( ˆ N n ) N n ˆ N n = {[a] a N n a(n)} 7
8 Nˆ n = {[a] a d([a], ˆ [0]) < 1 n } ˆM ˆM { Nˆ n } N n ˆM [a] N n [a] 1 n [a] + ˆ N n 1.4 R {I n } N n R I 1 I 2 I 3 I n N n I n = {0 R } R ˆR R a b ab (ab)(i) = a(i)b(i) i = 1, 2,, n, ab R a b ab ˆR [a] [b] [a][b] [ab] Well-defined () (1) R (R ) (2) ˆR (3) N n Î n {[a] a R I n Î n ˆR 1 n Î n ˆR (4) ˆR a(n)} 8
9 2 Z 2.1 s 1, s 2,, s m, n s n s n+1 lim s n = n Z {s n Z} N n s 1 Z s 2 Z s 3 Z s n Z N n s n Z = {0} ( ) a a(i) = a(i + 1) Mod s i i = 1, 2, 3,, n, a ( (Z, {s n Z} N n ) ) (Z, {s n Z} N n ) (1) a n, m n < m a(n) = a(m) Mod s n (2) a, b a = b (3) a b b(n) = a(n) Mod s n b a (4) a a (1) a n, m n < m a(n) = a(n + 1) Mod s n 0 a(n) < s n s n Z a(n + 1) a(n) s n+1 Z a(n + 2) a(n + 1), s n+2 Z a(n + 3) a(n + 2),.. s m 1 Z a(m) a(n 1) s n Z s n+1 Z s n+2 Z s m 1 Z s n Z a(m) a(n) 0 a(n) < s n 9
10 a(n) = a(m) Mod s n (2) a, b n a, b n m n < m a(m) b(m) (mod s n ) (1) a(n) = a(m) Mod s n b(n) = b(m) Mod s n a(n) = b(n) a = b (3) n a a(n) a(n + 1) (mod s n ) b(n + 1) = a(n + 1) Mod s n+1 b(n + 1) a(n + 1) (mod s n ) a(n) b(n + 1) (mod s n ) b(n) = a(n) Mod s n = b(n + 1) Mod s n b m n b(m) a(m) (mod s m ) c n c m b(m) a(m) (mod s n ) b a (4) a a a (2) a a N(a) (4) ( ) (1) a n N 10
11 N m m a(n) a(m) (mod s n ) N(a)(n) = a(n) Mod s n (2) a N(a)(n) = a(n) Mod s n () m 1, m 2,, m n, a m1, a m2,, a mn, a mi = a mi+1 Mod s mi (i = 1, 2, ) a a(n) = a mk Mod s n ( n m k k ) a a m1, a m2,, a mn, a(m i ) = a mi (i = 1, 2, ) {a mi } N i 2.2 Z p p Z {p n Z} N n Z Z p (Z, {p n Z} N n ) a Z p [a] p a N p (a) d p (Z Z p ) (Z, {p n Z} N n ) (1) a N n N N s.t. N m s.t. N < m p n a(n) a(m) 11
12 (2) a N n, m s.t. n < m p n a(n) a(m) (3) a N n a(n) = a(n + 1) Mod p n N n, m s.t. n < m a(n) = a(m) Mod p n 2.3 Ẑ Z {(n + 1)!Z} N n Z Ẑ (Z, {(n + 1)!Z} N n ) a Ẑ [a] a N(a) d (Z Ẑ ) (Z, {(n + 1)!Z} N n ) (1) a N n N N s.t. N m s.t. N < m (n + 1)! a(n) a(m) (2) a N n, m s.t. n < m (n + 1)! a(n) a(m) (3) a N n a(n) = a(n + 1) Mod (n + 1)! N n, m s.t. n < m a(n) = a(m) Mod (n + 1)! 2.4 Ẑ Z p (n + 1)! (φ p (n)) p n φ p (n) = max{m Z p m (n + 1)! } φ p (n) n (1) p φ p (n) 12
13 (2) (n + 1)! = p prime φ p (n)>0 p φ p(n) (n + 1)! p Ẑ Z p p a (Z, {(n + 1)!Z} N n ) a (Z, {p n Z} N n ) (2) a (Z, {(n + 1)!Z} N n ) 0 a (Z, {p n Z} N n ) 0 (3) Ẑ Z p Ẑ [a] Ẑ p [a] p Well-defined ϕ p ϕ p : Ẑ Z p ϕ p ([a]) = [a] p (1) n n (n + 1)! p n (n = p n ) a (Z, {(n + 1)!Z} N n ) N N m m (n + 1)!Z a(m) a(n) (n + 1)! p n p n Z a(m) a(n) a (Z, {p n Z} N n ) (2) n n (n + 1)! p n a (Z, {(n + 1)!Z} N n ) 0 N N m m (n + 1)!Z a(m) (n + 1)! p n p n Z a(m) a (Z, {p n Z} N n ) 0 (3) ϕ p well-defined (1), (2) ϕ p a (Z, {(n + 1)!Z} N n ) ϕ p ([a]) = [a] p = [N p (a)] p N p (a) a (Z, {p n Z} N n ) 13
14 N p (a)(n) = a(m) Mod p n ( m n φ p (m) ) m N p (a)(φ p (m)) = a(m) Mod p φ p(m) (Ẑ Z p ) : Ẑ p prime ϕ p p prime Z p Chinese Remainder Theorem (Chinese Remainder Theorem) m n Z/mnZ Z/mZ Z/nZ ( x + mnz (x + mz, x + nz) ) m n 0 a < m, 0 b < n 0 c < mn c c + mz = a + mz, c + nz = b + nz Chinese Remainder Theorem (Chinese Remainder Theorem) m = p e 1 1 pe 2 2 pe s s m 0 a 1 < p e 1 1, 0 a 2 < p e 2 2,, 0 a s < p e s s a 1, a 2,, a s 0 a < m a a + p ei i Z = a i + p ei Z (i = 1, 2,, s) i Ẑ [a] ( ϕ p )([a]) p prime p ϕ p ([a]) = [0] p a (Z, {(n + 1)!Z} N n ) 14
15 p [0] p = ϕ p ([a]) = [N p (a)] p N p (a) (Z, {p n Z} N n ) N p (a) = 0 n (n + 1)! = p prime φ p >0 p φ p(n) (n + 1)! 0 = N p (a)(φ p (n)) = a(n) Mod p φ p(n) 0 a(n) < (n + 1)! a(n) = 0 ( ϕ p ) p prime a p p prime Z p ([a p ] p ) (Z, {p n Z} N n ) n (n + 1)! = p prime φ p >0 p φ p(n) (n + 1)! x + p φp(n) Z = a p (φ p (n)) + p φp(n) Z ( p prime φ p > 0) 0 x < (n + 1)! a(n) a a (Z, {(n + 1)!Z} N n ) ( ϕ p )([a]) = ([a p ] p ) p prime a p φ p (n) > 0 a(n + 1) Mod p φ p(n+1) = a p (p φp(n+1) ) a(n+1) Mod p φ p(n) = a p (φ p (n+1)) Mod p φ p(n) = a p (φ p (n)) = a(n) Mod p φ p(n) Chinese Remainder Theorem a(n + 1) Mod (n + 1)! = a(n) 15
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