课程名称︰分析导论一
课程性质︰数学系大二必修兼经济系、经研所选修
课程教师︰陈俊全 教授
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2018/10/30
考试时限（分钟）：09:20~11:05 共105分钟
试题 :
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