Κεφάλαιο 1 Πραγματικοί Αριθμοί 1.1 Σύνολα

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3 x = 0

4 N = {1, 2, 3....}, Z Q a, b a, b N c, d c, d N a + b = c, a b = d. a a N 1 a = a 1 = a. < > P n P (n) P (1) n = 1 P (n) P (n + 1) n n + 1 P (n) n P (n) n P n P (n) P (m) P (n) n m P (n + 1)

5 P (n) n m P n P (n) P (1) P (1), P (2),..., P (n) P (n + 1) P (n) n n 1 P (n) : n k = n = k=1 n(n + 1). 2 1(1 + 1) P (1) : 1 =, 2 n = 1 n P (n) n + 1 P (n + 1) : (n + 1) = (n + 1)(n + 2) (n + 1) = n + (n + 1) n(n + 1) = + (n + 1) 2 ( n ) = (n + 1) = (n + 1)(n + 2). 2 a + x = b, a, b N, a < b a = b a > b Z = {0, ±1, ±2, ±3,...}.

6 a 0 + a = a + 0 = a, a a a + ( a) = ( a) + a = 0. < > a a = 2b, b Z a = 2b + 1, b Z a, b b 0 bx = a b a b a Q = { m n : m Z, n N }. m 1 n 1 = m 2 n 2 m 1 n 2 = n 1 m 2, m 1 + m 2 = m 1n 2 + m 2 n 1 m 1, m2 = m 1m 2, n 1 n 2 n 1 n 2 n 1 n 2 n 1 n 2 m 1 > m 2 m 1 n 2 > m 2 n 1 m 1 n 2 m 2 n 1 > 0, m 1 n 2 m 2 n 1 N. n 1 n 2 q q = m n m n

7 2 6 = 1 3, = 14 8 = x x 2 = 2. q Q q 2 = 2 q = a a b b q 2 = 2 q 2 = 2 a2 b 2 = 2 a2 = 2b 2. a 2 a a = 2c c Z 4c 2 = 2b 2 2c 2 = b 2. b 2 b a b x 2 = L R

8 l L r R l < r L R l 2 1, 1.4, 1.41, 1.414, ,..., r 2 l 2, 1.5, 1.42, 1.415, ,.... a 1, 1.4, 1.41, 1.414,... a a 2, 1.5, 1.42, 1.415,... a 2 L R 2 R Q A = A R A a R x a x A a A

9 A a R x a x A a A A 1 2, 2 3, 3 4,..., n n + 1,..., n N. 1 2 x 1 x 3 x 1 < x < 3 = A R a A a 1 R a 1 a a 1 a A x a a 1 a 1 a x x A A a 1 a A a R A a A a 1 R A a a 1 a R A a A

10 a 1 R A a a 1 A A A A A A A A A A A A A a, a 1 A a a 1 a 1 a a = a 1 A 1 2, 2 3, 3 4,..., n n + 1,..., n N. 1 a < 1 2 n n > 1 n + 1 > a x 1 x 3 x 1 < x < 3 A = {x Q : x > 0 x 2 < 2} A = 0 A = 2

11 2 A = {x Q : x > 0 x 2 < 2} A Q A = 2 Q A A = 2 2 A R A R 2 R = {a : a }. a, b c a + b = c R 1 a, b, c R (a + b) + c = a + (b + c) ( ). R 2 a, b R a + b = b + a ( ).

12 R 3 R a R a + 0 = 0 + a = a ( ). R 4 a R R a a a + ( a) = ( a) + a = 0 ( ). a, b R a b a b a b = a + ( b). a, b c a b = c R 5 a, b, c R (a b) c = a (b c) ( ). R 6 a, b R a b = b a ( ). R 7 R a R a 1 = 1 a = a ( ). R 8 a R, a 0 R a 1 a a a 1 = a 1 a = 1 ( ). a, b R, b 0 a b a b a b = a b 1.

13 R 9 a, b, c R a (b + c) = a b + a c ( ). R < R 10 a, b R a < b a = b a > b ( ). R 11 a, b, c R a < b b < c a < c R 12 a, b, c R a < b a + c < b + c R 13 a, b, c R a < b c > 0 a c < b c x R a R a + x = a 0 + x = 0. R 3 x + 0 = x. R 2 x = x + 0 = 0 + x = 0. a R b, c R a + b = a + c = 0 R 3 R 1 R 2 R 3 c = c + 0 = c + (a + b) = (c + a) + b = (a + c) + b = 0 + b = b. R 1 R 8 R R 1 R 4 R R 5 R 8 R R 1 R 9

14 R 10 R 13 (R, +,, <) (Q, +,, <) x 2 = 2, x R x 2 = 2 R A R y A y a a A a y a A y A = { a : a A} A ( A) = s t = s A s a a A t = s a a A t A t 1 A t 1 a a A a t 1 a A t 1 A s t 1 t 1 s = t A R a A a = A > 0 x A x > a a A a = A > 0 x A x < a +

15 a = A > 0 x A x a a A a a, 0, > 0 x A x > a A = b < a = a b > 0 b A x A x b = a, a = A x, y R x > 0 n N nx > y nx y n N A = {nx : n N} A A = s x > 0 s x < s n 0 N n 0 x > s x s < (n 0 + 1)x s A x R a Z a x < a + 1 a x [x] [x] x < [x] [2.4] = 2, [ 4.7] = 5. x, y R x < y q Q x < q < y

16 y x > 0 n N n(y x) > 1 nx + 1 < ny nx < [nx] + 1 nx + 1 < ny, x < [nx] + 1 < y, n q = [nx] + 1 n x, y R x < y p x < p < y x < y x 2 < y 2 q Q x 2 < q < y 2, x < q + 2 < y. p = q + 2 a R { a a 0, a = a a < 0. a a a a R b 0 a b b a b.

17 a, b R a + b a + b ( ). a b a b a b a + b ( ). a 0 a < 0 a a a b b b, ( a + b ) a + b a + b, a = a b + b a b + b a b a b. b = b a + a b a + a = a b + a a b a b. a b a b a b, b b b = b + < a a < + a R. + +

18 a R a + ( ) = ( ) + a = a (+ ) =, a + (+ ) = (+ ) + a = a ( ) = +. a > 0 a ( ) = ( ) a =, a (+ ) = (+ ) a = +. a < 0 a ( ) = ( ) a = +, a (+ ) = (+ ) a =. ( ) + ( ) =, (+ ) + (+ ) = +, ( ) ( ) = +, (+ ) (+ ) = +, ( ) (+ ) = (+ ) ( ) =. ( ) + (+ ), (+ ) + ( ), 0 ( ), ( ) 0, 0 (+ ), (+ ) 0,, +, +, + +. A 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 : x > a}, [a, ) = {x R : x a}, (, a) = {x R : x < a}, (, a] = {x R : x a}.

19 n 2k k=1 n (2k 1). k=1 n k 2 = k=1 n(n + 1)(2n + 1). 6 n (2k) 2 k=1 n (2k 1) 2. k=1 n (2k + 1)2 k = A + (B + Cn)2 n, k=0 A, B C n n (i) 2 n > n 2 (ii) 2 n > n 3 x > 1 (1 + x) n 1 + nx n N q Q q 3 = p, q, r, s Q p + q = r + s, p = r, q = s q s A = {x R : x > 0 0 < x 2 1 2}

20 B = {x Q : x 0 0 < x 2 1 2} C = {0, 1, 1, 1,..., 1,...} n D = {x R : x < 0 x 2 + x 1 < 0} E = { 1 + n ( 1)n : n N} A R A = A A A, B R A B B A A B. A, B R A+B = {a+b : a A, b B} (A + B) = A + B. R x < y + > 0 x y x y + > 0 x y x y > 0 x = y a < x < b a < y < b x y < b a

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