课程名称︰分析导论一
课程性质︰
课程教师︰陈俊全
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2018/11/13
考试时限（分钟）：3 小时
试题 :
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.