1 I

1 I

3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y = y x (1.7) 1( 0) R : 1 x = x x R (1.8) x R : x 0 R, 1 x 1 R : x x 1 = 1 y 0 x y 1 x/y (1.9) a (x + y) = a x + a y x + ( y) x y, x y xy x, y R x < y, x = y, x > y x < y x = y x y (2.1) x y, y z = x z. (2.2) x y = a + x a + y a R. (2.3) x y, a 0 = ax ay. R (A, B) max A min B

4 1 1.2 A R 1.2.1. A (bounded from above) α R x A x α ( α R : x α x A.) A (bounded from below) β R x A x β ( β R : x β x A.) A A (bounded) ( α, β R : β x α x A.) α A (upper bound) β A (lower bound) α A α α α A A, A a, b R a < b [a, b] = {x R; a x b} : (a, b) = {x R; a < x < b} : (a, b] = {x R; a < x b} : [a, b) = {x R; a x < b} : [a, ) = {x R; a x} : (, b] = {x R; x b} : (a, ) = {x R; a < x} : (, b) = {x R; x < b} : (, ) = R : 1.2.2. (1) A = (0, 1] A 1 1.5 A 1 0

1.2. 5 (2) A = { 1 n } n=1,2,... = {1, 1/2, 1/3, 1/4,...} A 1 1 (3) A = [1, ) 0 (4) A = {n} n=1,2,3,... = {1, 2, 3,...} 1 1.2.3. A α α A A (maximum) α = max A { x α, x A α = max A α A A β β A A (minimum) β = min A { x β, x A β = min A β A 1.2.4. (1) A = [0, 1] max A = 1, min A = 0. (2) A = {1, 1/2, 1/3,...} max A = 1, min A (3) A = [1, ) min A = 1, max A (4) A = {1, 2, 3,...} min A = 1, max A 1.2.5. R A, B (A, B) R (cut) (1) A, B. (2) A B = R. (3) A B =. (4) x A, y B, x < y. (Dedekind) R (A, B) max A min B

6 1 1.2.6. (A, B) max A, min B (A, B) α = max A, β = min B α A, β B (4) α < β α < (α+β)/2 < β (α+β)/2 R (2) (α+β)/2 A (α+β)/2 B ( (α+β)/2 A α = max A 1.2.7. A A B α A (supremum) α = sup A A sup A = A A B β A (infimum) β = inf A A inf A = 1.2.8. (1) A = [0, 1] sup A = 1, inf A = 0. (2) A = {1, 1/2, 1/3,...} sup A = 1, inf A = 0. (3) A = [1, ) inf A = 1, sup A = +. (4) A = {1, 2, 3,...} inf A = 1, sup A = +. (5) A = (1, 2] [3, 5) sup A = 5, inf A = 1. 1.2.9 ( (Weierstrass) ). E E B A = R \ B (A, B) R E B E a E a 1 E a 1 A A, B. A, B A B = R, A B = x A, y B x E z E : x < z.

1.3. 7 y E z y x < z y, x < y. (A, B) R max A min B α = max A α A α E z E : α < z. (α + z)/2 < z (α + z)/2 E (α+z)/2 A. α = max A min B = sup E 1.2.10. F F = { x; x F } sup( F ) = inf F 1.2.11. A (1) A { (i) x α, x A α = sup A (ii) ε > 0, x A : α ε < x (i) α A (ii) α A (2) A β = inf A { (i) x β, x A (ii) ε > 0, x A : β + ε > x 1.3 (sequence) a 1, a 2, a 3,... {a n } {a n } n=1 1.3.1. {a n } a a a {a n } ε > 0, N N : n N = a n a < ε.

8 1 ε > 0 N n N n a n a < ε For any ε > 0, there exists N N such that for all n N, a n a < ε. lim a n = a a n a (n ) {a n } (divergent) M > 0, N N : n N = a n M {a n } lim a n = + M > 0, N N : n N = a n M {a n } lim a n = 1.3.2. c a n = c (n = 1, 2,...) ε > 0 N N = 1 n N = 1 n a n c = c c = 0 < ε lim c = c 1.3.3. a n = 1/n ε > 0 N N = [1/ε] + 1 n N n 1/n < ε 0 < ε 1 n 1 lim n = 0

1.3. 9 x [x] x (Gauss) x [3] = 3, [3.1] = 3, [1/3] = 0. x 1 < [x] x. 1 lim = 0 n ε = 1/10 N N = 11 11 n 1 0 < 1/10 n ε = 1/100 N N = 101 101 n 1 0 < 1/100 n ε = 1/1000 N N = 1001 1001 n 1 0 < 1/1000 n ε > 0 N n a n a < ε ε > 0 N = [ 1 ] + 1 ε 2 lim 1 n = 0 1.3.4. {a n } a, b ε > 0 N 1 N : n N 1 = a n a < ε/2, N 2 N : n N 2 = a n b < ε/2 N = max(n 1, N 2 ) N N 1, N N 2 a N a < ε/2, a N b < ε/2 a = b a b a a N + a N b < ε.

10 1 1.3.5. {a n }, {b n } a, b (i) {a n + b n } a + b lim (a n + b n ) = a + b. (ii) {a n b n } ab lim (a nb n ) = ab. c {ca n } ca lim (ca n) = ca. (iii) b 0 {a n /b n } a/b a n lim = a b n b. N n b n 0 {a n /b n } n=n (iv) a n b n (n = 1, 2,...) a b. (v) a n c n b n (n = 1, 2,...) a = b {c n } lim c n = a (v) a = b ε > 0, N 1 N : n N 1 = a n a < ε, N 2 N : n N 2 = b n a < ε. N = max(n 1, N 2 ) n N c n a < ε. a ε < a n c n b n < a + ε.

1.3. 11 1.3.6. (i) 1 lim n = lim 1 2 n lim 1 n = 0. (ii) [ 1 + n 1 lim = lim n 2 n + 1 ] 2 n 1 = lim n + lim 2 1 n = 0. (iii) lim n n = 1. a n = n n 1 a n 0 n n = 1 + a n n n = (1 + a n ) n = 1 + n C 1 a n + n C 2 a 2 n + n(n 1) a 2 2 n. 2 0 a n n 1. 2 n lim = 0 (v) lim a n = 0. lim n 1 n n = lim (a n + 1) = lim a n + 1 = 1. 1.3.7. (1) A α = sup A x n A α = lim x n {x n } (2) A β = inf A x n A β = lim x n {x n } (1) {x n } α = sup A (2) {x n } β = inf A (1). n 1.2.9 α 1/n < x n α x n A 1.3.6 (v) {x n } lim x n = α 1.3.8.

12 1 1.3.9. {a n } {a n } ε = 1 N N : n N = a n a < ε = 1 a b a b a n < a + 1 (n N). M = max{ a 1, a 2,..., a N 1, a + 1} a n M (n = 1, 2,...) {a n } 1.3.10. {a n } n=1 {a n} n 1 < n 2 < {a nk } k=1 1.3.11. a a {a n } a ε > 0, N N; n N = a n a < ε {a nk } k=1 {a n} n 1 < n 2 < K N n K N k K a nk a < ε. 1.3.12. {a n } n=1 a 1 a 2 a 1 a 2 1.3.13. {a n } lim a n = sup{a n ; n = 1, 2,...} {a n } lim a n = inf{a n ; n = 1, 2,...}

1.3. 13 α = sup{a n ; n = 1, 2,...} 1.2.11 ε > 0 α ε < a N a N n N {a n } α {a n } α ε < a N a N+1 α n N a n α < ε. lim a n = α 1.3.14 ( (Napier) ). {a n = ( 1 + 1 n) n} e ( 1 ) n. e = lim 1 + n a n = 1 + n ( 1 ) n(n 1) ( 1 ) 2 n(n 1) 2 1( 1 ) n + + + n 2! n n! n = 1 + 1 + 1 ( 1 ) 1 ( 1 ) ( n 1) 1 + + 1 1. 2! n n! n n a n+1 = 1 + 1 + 1 ( 1 ) 1 ( 1 ) ( n 1) 1 + + 1 1 2! n + 1 n! n + 1 n + 1 + 1 ( 1 ) ( n ) 1 1. (n + 1)! n + 1 n + 1 a n < a n+1 {a n } n! 2 n 1 a n 1 + 1 + 2 1 + + 2 (n 1) < 2 + 2 1 1 1/2 = 3. {a n } 1.3.13 lim a n 1.3.15 ( (Archimedes) ). a, b > 0, n N : na > b. a, b > 0 n na b n b/a N α [α] + 1 > α [α] + 1 α = sup N

14 1 1.3.16 ( ). a < b = p Q : a < p < b. Q a < 0 < b 0 0 a < b b a(> 0) 1 1.3.15 n N : n(b a) > 1. 1 na 1.3.15 n N : n > na. n m m 1 na < m m n m 1 n a < m n = m 1 n + 1 n a + 1 n < b. a < b 0 0 b < a b < q < a q 1.3.17 (). [a n, b n ] (n = 1, 2,...) [a 1, b 1 ] [a 2, b 2 ] n=1[a n, b n ] (b n a n ) 0 ) n=1[a n, b n ] (n a 1 a 2 a n b n b 2 b 1 {a n } {b n } 1.3.13 a = lim a n = sup{a n ; n = 1, 2,...}, b = lim b n = inf{b n ; n = 1, 2,...}. n a n a, b b n n=1[a n, b n ] a, b. n=1[a n, b n ] b n a n 0 (n ) a = b n=1[a n, b n ] = {a}

1.3. 15 1.3.18. I n = (0, 1/n) (n = 1, 2,...) I 1 I 2 1/n 0 (n ) n=1i n = 1.3.19. (i) (ii) (iii) (iv) (i) = (ii) = (iii) = (iv) (iv) = (i) (A, B) R a A, b B a < b (a + b)/2 A a 1 = (a + b)/2, b 1 = b (a + b)/2 B a 1 = a, b 1 = (a + b)/2 [a 1, b 1 ] (a 1 + b 1 )/2 A a 2 = (a 1 + b 1 )/2, b 2 = b 1 (a 1 + b 1 )/2 B a 2 = a 1, b 2 = (a 1 + b 1 )/2 [a 2, b 2 ] [a n, b n ] [a 1, b 1 ] [a 2, b 2 ], b n a n = (b a)/2 n, a n A, b n B b n a n 0 (n ) α R n=1[a n, b n ] = {α} α A α B α A α < β β R ε = β α (> 0) b n a n 0 n b n a n < ε. a n α b n α b n < a n + ε α + ε = β. β B α = max A α B α = min B

16 1 1.4 1.4.1. {a n } a {a n } {a n } a 1.4.2. {a n } a {a n } a {1, 1, 1, 1,...} 1, 1 1.4.3 ( (Bolzano-Weierstrass)). {a n } c < d c a n d (n = 1, 2,...) [c, (c + d)/2], [(c + d)/2, d] a n [c 1, d 1 ] [c 1, (c 1 + d 1 )/2], [(c 1 + d 1 )/2, d 1 ] a n [c 2, d 2 ] [c n, d n ] [c 1, d 1 ] [c 2, d 2 ] [c n, d n ], d n c n 0 (n ) α n=1[c n, d n ] = {α} [c 1, d 1 ] a n1 [c 2, d 2 ] n 1 < n 2 a n2 [c 2, d 2 ] n 1 < n 2 <, a nk [c k, d k ] (k = 1, 2,...) c k a nk d k, lim c k = α, lim d k = α k k {a nk } k=1 α 1.4.4. {a n } ε > 0 N N n, m N n, m a n a m < ε ε > 0, N N : n, m N = a n a m < ε.

1.4. 17 1.4.5. {a n } a ε > 0, N N : n N = a n a < ε/2. n, m N a n a m a n a + a a m < ε/2 + ε/2 = ε. {a n } 1.4.6. {a n } 0 a < 1 a n+1 a n a a n a n 1, n = 2, 3,... {a n } a n+1 a n a a n a n 1 a 2 a n 1 a n 2 a n 1 a 2 a 1 n > m a n a m a n a n 1 + a n 1 a n 2 + a n 2 a n 3 + + a m+1 a m (a n 2 + a n 3 + a m 1 ) a 2 a 1 am 1 (1 a n m ) a 2 a 1 1 a am 1 1 a a 2 a 1. 0 a < 1 ε > 0 N a N 1 1 a a 2 a 1 < ε n > m N a n a m < ε

18 1 1.4.7. A R A {a n } a A lim a n = a 1.4.8 ( ). R R R {a n } Step 1. {a n } ε = 1 N N : n N = a n a N < 1 a n a n a N + a N 1 + a N ( n N). M = max{ a 1,..., a n 1, a N + 1} a n M (n = 1, 2,...) {a n } Step 2. {a n } ε > 0, N N : n, m N = a n a m < ε/2. Step 1 {a n } Bolzano-Weierstrass {a nk } k=1 lim k a nk = a ε > 0, K N : k K = a nk a < ε/2. n K N n N a n a a n a nk + a nk a < ε/2 + ε/2 = ε. lim a n = a 1.4.9. R R \ {a} {a + 1/n} R \ {a} R \ {a}

19 2 2.1 2.1.1. (c, d) a (c, d) (c, d) \ {a} f(x) (1) A ε > 0, δ > 0 : 0 < x a < δ = f(x) A < ε f(x) x a A (2.1.1) lim x a f(x) = A (2) f(x) A (x a) M > 0, δ > 0 : 0 < x a < δ = f(x) M lim x a f(x) = M > 0, δ > 0 : 0 < x a < δ = f(x) M lim x a f(x) = (3) A ε > 0, M > 0 : x > M = f(x) A < ε lim x f(x) = A lim x f(x) = A, lim x ± f(x) = ±

20 2 2.1.2. (2.1.1) lim x a x a a 2.1.3. (1) f(x) = x + b lim x a f(x) = a + b. ε > 0 δ = ε 0 < x a < δ (2) lim x 1 x 2 1 x 1 f(x) (a + b) = (x + b) (a + b) = x a < δ = ε = 2. x 1 x 2 1 x 1 = (x 1)(x + 1) x 1 = x + 1 (1) lim x 1 x 2 1 x 1 = 2 2.1.4. lim x a f(x) = A x n a lim x n = a {x n } lim f(x n ) = A lim x a f(x) = A ε > 0, δ > 0 : 0 < x a < δ = f(x) A < ε. x n a lim x n = a {x n } N N : n N = 0 < x n a < δ f(x n ) A < ε lim f(x n ) = A. lim x a f(x) = A ε > 0 : δ > 0 : x : 0 < x a < δ, f(x) A ε. δ = 1/n x n 0 < x n a < 1/n, f(x n ) A ε. lim x n = a, (x n a) lim f(x n ) = A

2.1. 21 2.1.5. f(x), g(x) x a A, B lim x a f(x) = A, lim x a g(x) = B (i) lim x a (f(x) + g(x)) = A + B. (ii) lim x a cf(x) = ca. c (iii) lim x a f(x)g(x) = AB. (iv) B 0 lim x a f(x)/g(x) = A/B. (v) f(x) g(x), (x (c, d) \ {a}) A B. 2.1.4 1.3.5 2.1.6 ( ). (c, d) a (c, d) (c, d) \ {a} f(x) A ε > 0, δ > 0 : 0 < x a < δ = f(x) A < ε f(x) x a + 0 A lim f(x) = A f(x) A (x a + 0) x a+0 ε > 0, δ > 0 : δ < x a < 0 = f(x) A < ε f(x) x a 0 A lim f(x) = A f(x) A (x a 0) x a 0 2.1.7. f(x) = [x] lim x 1+0 f(x) = 1, lim x 1 0 f(x) = 0 2.1.8. lim x a f(x) = A lim x a+0 f(x), lim x a 0 f(x) lim x a+0 f(x) = lim x a 0 f(x) = A

22 2 2.2 I f(x) a I 2.2.1. (1) f(x) x = a (continuous) ε > 0, δ > 0 : x a < δ (x I) = f(x) f(a) < ε (2) f(x) I I 2.2.2. f(x) x = a x a A f(a) lim x a f(x) = f(a) f(x) x = a f : cont. at x = a, I f : cont. in I 2.2.3. lim x a+0 f(x) = f(a) f(x) x = a lim x a 0 f(x) = f(a) x = a f(x) x = a x = a 2.2.4. (1) f(x) = c R a R ε > 0 δ > 0 x a < δ = f(x) f(a) = c c = 0 < ε. (2) f(x) = x R a R ε > 0 δ = ε(> 0) x a < δ = f(x) f(a) = x a < δ = ε. 2.2.5. f(x) I a f(a) 0 x = a x f(x) f(a) δ > 0 : x a < δ = f(x)f(a) > 0. f(a) > 0 ε = f(a)(> 0) δ > 0 : x a < δ = f(x) f(a) < ε. f(a) ε = 0 < f(x)

2.2. 23 2.2.6. f(x) x = a lim x n = a lim f(x n ) = A 2.1.4 2.2.7. (i) f(x), g(x) I x = a I f(x) + g(x), cf(x) (c ), f(x)g(x), f(x)/g(x) (g(a) 0 ) x = a (ii) y = f(x) x = a z = g(y) y = f(a) g(f(x)) x = a 2.1.5 2.2.8. f(x) = x 2 (= x x) R f(x) = x n (n = 1, 2,...) R (1), (2) (1) P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 R. (2) R(x) = P (x)/q(x) P (x), Q(x) Q(x) 0 x (3) f(x) = sin x, f(x) = cos x R f(x) = tan x cos x 0 sin x x ( sin x sin y = x + y sin + x y ) ( x + y sin x y ) 2 2 2 2 = 2 cos x + y sin x y 2 2 2 sin x y 2 x y x = a ε > 0 δ = ε x a < δ sin x sin a x a < δ = ε sin x x = a cos x = sin(π/2 x) cos x R tan x = sin x/ cos x R \ {x R; cos x = 0}

24 2 (4) f(x) = e x ( e x = lim 1 + x n. n) e x R e x > 0, e 0 = 1. e x+y = e x e y. x < y = e x < e y. 2.2.9 ( ). f(x) I = [a, b] f(x) I α = sup{f(x); x I} α = +. x n I α = lim f(x n ) {x n } {x n } {x nk } k=1 c a x n k b a c b c I = [a, b]. f(x) x = c f(c) = lim k f(x nk ) = α α f 2.2.10. [a, ) f(x) = x (0, 1) f(x) = x (0, 1) f(x) = 1 x 2.2.11 ( ). f(x) I = [a, b] α = f(a), β = f(b) α β α β γ (γ α, β) f(c) = γ c (a, b) 1 f α < β α < γ < β E = {x I; f(x) < γ} f(a) < γ E

2.2. 25 c = sup E 2.2.5 c (a, b) x n E : x n c (n ). f(x) f(c) = lim f(x n ) γ. f(c) < γ 2.2.5 c f(x) < γ c f(c) = γ 2.2.12. 2.2.9 2.2.11 I = [a, b] f(x) α = min x I f(x), β = max x I f(x) f([a, b]) = [α, β] α = β [α, β] = {α} 2.2.13. I f(x) x 1, x 2 I, x 1 < x 2 f(x 1 ) < f(x 2 ) f(x) I x 1, x 2 I, x 1 < x 2 = f(x 1 ) > f(x 2 ) 2.2.14. y = f(x) I = [a, b] α = f(a), β = f(b) α = f(b), β = f(a) f(x) x = f 1 (y) [α, β] f(x) [a, b] f([a, b]) = [α, β] f(x) x = f 1 (y) f 1 (y) y 1 < y 2 y 1 = f(x 1 ), y 2 = f(x 2 ) x 1, x 2 [a, b] x 1 = f 1 (y 1 ), x 2 = f 1 (y 2 ) x 1 x 2 f(x) (y 1 =)f(x 1 ) (y 2 = )f(x 2 ) y 1 < y 2 x 1 < x 2 f 1 (y 1 ) < f 1 (y 2 ). f 1 (y) [α, β]

26 2 f 1 (y) y 0 [α, β] (1) ε 0 > 0 : δ > 0, y [α, β] : y y 0 < δ, f 1 (y) f 1 (y 0 ) ε 0. δ = 1/n y n y n = f(x n ) x n [a, b] y n y 0 < 1/n, f 1 (y n ) f 1 (y 0 ) ε 0 {x n } {x nk } k=1 lim k x nk = x 0 x 0 [a, b] f(x) [a, b] lim k f(x nk ) = f(x 0 ). y n y 0 y 0 = f(x 0 ) f 1 (y nk ) f 1 (y 0 ) = x nk x 0 0 (n ) (1) f 1 (y) [α, β] 2.2.15. a < b f(x) I = (a, b) f((a, b)) = (α, β) y = f(x) x = f 1 (y) (α, β) f(x) [c, d] (a, b) 2.2.14 2.2.16. y = f(x) : [a, b] [α, β] x = f 1 (y) : [α, β] [a, b] x y = f 1 (x) y = f 1 (x), x [α, β] x = f(y), y [a, b]. y = x 2.2.17. y = x n/m n, m N n/m m y = x m : [0, ) [0, ) y = x 1/m : [0, ) [0, ) m y = x m : (, ) (, ) y = x 1/m : (, ) (, ) y = x n/m = (x 1/m ) n y = x 1/2 = x [0, ) y = x 1/3 = 3 x (, )

2.2. 27 2.2.18. f(x) = log x y = e x y = log x, (x > 0) x = e y f(x) = log x x > 0 y = log x : (0, ) R y = e x : R (0, ) 2.2.19. a > 0 f(x) = a x a x = e x log a a > 1 a x 0 < a < 1 a x 2.2.20. (1) f(x) = sin x [ π/2, π/2] [ 1, 1] [ 1, 1] [ π/2, π/2] f 1 (x) = Sin 1 x y = Sin 1 x x = sin y, π/2 y π/2. (2) f(x) = cos x [0, π] [ 1, 1] [ 1, 1] [0, π] f 1 (x) = Cos 1 x y = Cos 1 x x = cos y, 0 y π. (3) f(x) = tan x ( π/2, π/2) (, ) (, ) ( π/2, π/2) f 1 (x) = Tan 1 x y = Tan 1 x x = tan y, π/2 < y < π/2. 2.2.21. I f(x) I ε > 0 δ > 0 x y < δ x, y I f(x) f(y) < ε ε > 0, δ > 0 : x y < δ = f(x) f(y) < ε.

28 2 2.2.22. I I ε > 0 δ > 0 f(x) I a I, ε > 0, δ > 0 : x a < δ = f(x) f(a) < ε. f(x) I ε > 0, δ > 0 : x a < δ = f(x) f(a) < ε. ε > 0 δ > 0 2.2.23. (1) f(x) = x (0, 1] (2) f(x) = x 2 R ε > 0 : δ > 0, x 1, x 2 R : x 1 x 2 < δ, f(x 1 ) f(x 2 ) ε. ε = 1 δ > 0 x 1 x 2 = δ/2, x 1 + x 2 > 2/δ x 1, x 2 R x 2 1 x 2 2 = x 1 x 2 x 1 + x 2 > δ 2 2 δ = 1. 2.2.24. f(x) I = [a, b] I f(x) I ε > 0 : δ > 0, x 1, x 2 I : x 1 x 2 < δ, f(x 1 ) f(x 2 ) ε. δ = 1/n x 1,n, x 2,n I (1) x 1,n x 2,n < 1/n, f(x 1,n ) f(x 2,n ) ε. {x 1,n } I {x 1,nk } lim k x 1,nk = x 0 I

2.2. 29 x 0 I x 1,nk x 2,nk < 1/n k lim k x 2,nk = x 0 f(x) lim f(x 1,n k ) = f(x 0 ) = lim f(x 2,nk ). k k f(x 1,nk ) f(x 2,nk ) 0 (k ) (1) 2.2.25. f(x) = x 2 I = [a, b] 2.2.24 R