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