作者David0620 (debeer)
看板NTU-Exam
標題[試題] 107-1 陳俊全 分析導論一 第一次小考
時間Tue Oct 30 19:33:41 2018
課程名稱︰分析導論一
課程性質︰數學系大二必修兼經濟系、經研所選修
課程教師︰陳俊全 教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2018/10/30
考試時限（分鐘）：09:20~11:05 共105分鐘
試題 :
There are 4 problems with a total of 80 points. Please explain all the
reasons in detail and show all of your work.
1. (10% + 10% = 20%)
If {x_n}, {y_n} are Cauchy sequence in Ｒ. Used ε-Ν to prove that
(a) {x_n + y_n} is also a Cauchy sequence.
(b) {x_n * y_n} is also a Cauchy sequence.
2. (20%)
If {x_n} is a sequence in Ｒ, and there exist a real number a ∈ (0, 1),
such that
|x_(n + 1) - x_n| <= a^n
Prove that x_n ---> x for some x in Ｒ.
3. (15% + 5% = 20%)
(a) Prove that for any rational number r the set r' = {q ∈ Ｑ: q < r} is a
Dedekind cut, and hence a real number.
(b) What cut (real number) does the following represent:
{q ∈ Ｑ: ∃ n ∈ Ｎ such that q <= (1 + (1 / n))^n}.
4. (10% + 10% = 20%)
If {x_n} is a sequence in Ｒ, satisfies:
i. 0 < x_1 < 1
ii. x_(n + 1) = 1 - (1 - x_n)^(1/2), ∀n >= 1
Prove that
(a) x_n ---> 0 as n ---> ∞.
(b) x_(n + 1) / x_n ---> 0.5 as n ---> ∞.
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