数学专业英语作业

作业
一、 汉译英
1．唯一性：若数列{n x }收敛，则它只有一个极限.
证明. 设lim ,n n x A →∞
= 对于给定的正数ε，存在正整数1
,N 对于一切 1,n N >有
||.2n x A ε-< （1） 又设lim ,n n x B →∞= 对于上述的正数ε，存在正整数2,N 对于一切2,n N >有
||.2n x B ε-< （2） 取12max{,},N N N =当,n N >（1）与（2）式都成立，此时有
|||()()|||||,22
n n n n A B x B x A x A x B εεε-=---≤-+-<+=
即||,A B ε-<由于A ，B 均为常数，而ε是任意正数，故只能.A B =
2． 基于学生数学能力结构的分析提出的培养建议
摘要 公民数学素养的提高无论对于社会发展还是其个人发展都具有重要意义。

为了提高中小学数学教育质量，必须明确中小学数学能力结构。

中小学生的数学能力可以分为两个层次，运算能力、空间想象能力、信息处理能力是第一个层次，逻辑思维能力和问题解决能力是第二个层次；模式能力在这两个层次之间起着非常重要的桥梁作用。

据此，本文提出了中小学生数学能力培养的几点建议。

关键词 中小学生；数学能力结构；模式能力
二、英译汉
1．The Role of Mathematics in Economics. Economics is a mathematical discipline. This assertion may seem strange to the traditional political economist, but mathematical methods were introduced at an early stage (Cournot,1838) in the two-hundred-year history of our subject and have been steadily growing in significance. At the present time and, essentially, since the end of World War II, mathematical methods have become predominant in American economics. The mathematical approach was originally inspired in Europe and England but it has flowered in America, with no little stimulus from European immigrants. The mathematical approach is steadily gaining favor throughout the world, especially because the younger generation in developing economics is embracing the new methods and because the socialist countries have shed a previous bias against the use of mathematical methods in economics. It is clear that the future development of economics will see continued and increasing use of mathematics, although it would be rash to assume that the future course of economic analysis will be predominantly mathematical as it has been in the last twenty years.
2. The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it is
feasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.
3．The values of the constructed function should not exceed the maximum permissible.
4．Rational number can be represented by the quotient of two integers, which is different from irrational one.
5．It should be noted that equation (1) and (2) are merely two different ways of expressing precisely the same relations.。