确界和确界存在性的变式教学研究  

Study on Variant Teaching for Supremum and Infimun and their Existence Theorem

作　　者：方静[1] 刘静[1] 卢卫君[1] 

机构地区：[1]广西民族大学理学院,广西南宁530006

出　　处：《广西民族大学学报（自然科学版）》2017年第1期95-103,共9页Journal of Guangxi Minzu University ：Natural Science Edition

基　　金：广西教育厅高校科研项目(KY2015ZD038)

摘　　要：确界存在性定理反映了实数系连续性/完备性这一基本性质,是高等微积分中极限理论的基石,由它可以循环推证实数完备性的其他五个等价定理.然而,教学实践表明大学一年级初学者在接触上(下)确界的形式逻辑符号语言及其推理论证时,普遍感到异常抽象难以理解和运用,难以达到既定的教学目标.笔者针对这种现状,从直观可视化和层次化的角度去剖析上/下确界的概念;对于数集的确界存在性定理,结合确界的无穷小数逼近表示法和确界的集合分划理念进行变式教学,以期将复杂抽象的确界概念和确界存在性的论证思想尽可能的通俗简单化,使得初学者尽可能分享到看得懂、听得懂的益处.The existence theorem of exact upper/lower bound reflects a fundamental property of real 1 number system with continuity or completeness and becomes a cornerstone for limit theory in Advanced Dif- ferential Calculus, from which one can cyclically deduce the other five equivalent theorems on the complete- ness of real numbers. However, the teaching practice showed that the most beginners of college freshmen feel it very so abstract and hard to understand when they first contact the formal logical language of supre- mum or infimum , especially further use them to verify, such that the given teaching target is unaccessible. Based on current situation, this paper firstly aims at two directions： one is to dissect concepts of supremum and infimum from intuitive visual view and stratification mode; another is to perform a variant teaching to the existence theorem for exact upper/lower bound by combining the technique from infinite decimal approxi- mate fraction and partition of a set. The purpose of doing that is to expect that the abstract and complicated concept of exact bound and the reasoning idea for proving the existence of exact bound theorem are dealt with visualization and simplification, and that the beginners feel benefits with understanding to read and listen.

关 键 词：上/下有界 上/下界 上/下确界 确界存在性定理 无限小数逼近法 集合分划 

分 类 号：G642[文化科学—高等教育学] O171[文化科学—教育学]