課程名稱︰分析導論一 課程性質︰ 課程教師︰陳俊全 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2018/11/13 考試時限（分鐘）：3 小時 試題 : Choose 5 from the following 8 probelms. 1. Let (F, +, *, ≦) be an ordered field which satisfies the axioms: 1. (1) x + y = y + x. (2) x + (y + z) = (x + y) + z. (3) ∃0 s.t. 0 + x = x. (4) For eaxh x,∃-x s.t. x + (-x) = 0. 2. (5) x * y = y * x. (6) x * (y * z) = (x * y) * z. (7)∃1 s.t. 1 * x = x. (8) For each x ≠0, ∃x^-1 s.t. x * x^-1 = 1 (9)x * (y + z) = x * y + x * z.(10) 1 ≠ 0. 3. (11) x≦x. (12) If x≦y and y≦x, then x = y. (13)If x≦y and y≦z,then x≦z.(14) ∀x,y either x≦y, or y≦x. (15)If x≦y, then x + z≦y + z.(16) If 0≦x and 0≦y, then 0≦x*y. Use these axioms to prove the follwing properties: (a) If a + x = b + x, then a = b. (b) 0*x = 0 for every x. (c) -(-x) = x. (d) (-1) * x = -x. (e) 0 < 1. 2. Let A and B be two nonempty sets of real number with property that x≦y for all x∈A, y∈B. Show that there exists c∈R s.t. x≦c≦y for all x∈A, y∈B. 3. Assume x_n → x and y_n → y as n → ∞. Use the ε-N definition to prove the following statements: (a) Let z_n =x_n + y_{n+1} + x_{n+2} + y_{n+3}. Show that {z_n} is a Cauchy seq (b) Prove that x_n * y_n → x * y as n → ∞. 4. (a) For nonempty sets A,B⊂R, Let A + B={x + y : x∈A, y∈B}. Show that inf(A+B) = inf(A) + inf(B) (b) Let A,B⊂R be nonempty and f: A×B → R be bounded. Show that sup{f(x, y): (x, y)∈A×B} = sup{ sup{f(x, y): x∈A} : y∈B}. 5. Show that the following are equivalent for an order field. (a) Completeness Axiom:every nondecreasing bounded above sequence are converge (b) Least upper bounded property: Let S be a nonempty set that has an upper bound. Then S has a least upper bound. 6. n Let y_n = Σ 1/k and x_n = sin(y_n). Find limsup x_n, liminf x_n and k=1 n→∞ n→∞ the set of all cluster points of {x_n}. 7. (a) Let E = {(x, y): x*x + y*y < 3}⊂R×R. Show that E is an open set. (b) Let A⊂R be open and B⊂R. Sefine AB={xy∈R: x∈A, y∈B}. Show that AB is open if 0 not in B. 8. We say card A < card B if card A =/= card B and there exists a one to one function from A to B . Let {0, 1}^N = {(a_1, a_2, ..., a_k, ...) : a_k = 0 or 1 for k =1,2,3,...} E = {(a_1, a_2, ..., a_k, ...): a_k∈Z, 0≦a_k≦k for k=1,2,3,... } N^N = {(a_1, a_2, ...., a_k,...): a_k∈N for k =1,2,3,...} R^N = {(a_1, a_2, ...., a_k,...): a_k∈R for k =1,2,3,...} Is it true that card {0, 1}^N < card E < card N^N < card R^N ? Please compare the cardinalities of these sets. --

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