利用子列极限证明数列上下极限的性质
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摘要
本文在数列上下极限的几种等价定义中,选取了一种便于推导其性质的子列的最大最小极限定义作为出发点,证明了上下极限的计算公式以及包括四则运算在内的一系列性质.由此出发,还证明了函数在实数轴上连续且单调时,自变量数列与函数值数列上下极限关系以及它的一个逆命题,证明了数列斯托尔茨公式在上下极限的推广结果.最后,研究了将一个数列分成几个子列后,其上下极限与子列上下极限的关系.
In this paper, among several equivalent definitions of upper and lower limits of sequence, a definition of maximum and minimum limits of subsequences which is convenient to deduce their properties is chosen as the starting point, and the calculation formula of upper and lower limits and a series of properties including four operations are proved. From this point of view, we also prove the upper and lower limit relations between the sequence of independent variables and the sequence of function values and an inverse proposition of the function when the function is continuous and monotonous on the real number axis, and prove the upper and lower limit results of the generalization of Stolz's formula. After dividing a sequence into several subseguences, we study the relationship between the upper and lower limits of a sequence and those of subseguences.
In this paper, among several equivalent definitions of upper and lower limits of sequence, a definition of maximum and minimum limits of subsequences which is convenient to deduce their properties is chosen as the starting point, and the calculation formula of upper and lower limits and a series of properties including four operations are proved. From this point of view, we also prove the upper and lower limit relations between the sequence of independent variables and the sequence of function values and an inverse proposition of the function when the function is continuous and monotonous on the real number axis, and prove the upper and lower limit results of the generalization of Stolz's formula. After dividing a sequence into several subseguences, we study the relationship between the upper and lower limits of a sequence and those of subseguences.
引文
[1]芮绍平,张杰.上、下极限三种定义的等价证明[J].山西大同大学学报(自然科学版), 2009, 25(6):9-10, 43.
[2]隋廷芳.上下极限的七个等价定义[J].内蒙古电大学刊, 1994(S4):4-7.
[3]王岩岩,刘伟.数列上下极限的研究[J].赤峰学院学报(自然科学版), 2011, 27(1):1-2.
[4]王振福,张建军.数列的上极限与下极限探析[J].包头职业技术学院学报, 2008(1):27-28.
[5]吐尼亚孜·库比.数列上、下极限的初等性质[J].新疆教育学院学报(汉文综合版), 1998, 14(1):54-55.
[6]艾斯卡尔·阿布力米提.数列上、下极限性质的推广应用[J].新疆教育学院学报, 2003(4):92-94.
[7]楼红卫.上下极限视角下的斯托尔茨公式和罗必塔法则[J].大学数学, 2017 33(1):96-99.
[8]程其襄,张奠宙,魏国强等编.实变函数与泛函数分析基础[M].3版.北京:高等教育出版社,2018.
[9]余国林,魏本成.关于上、下极限的一个新定理[J].大学数学, 2007(5):163-166.
[10]卢志康.上下极限的概念在极限教学中的作用[J].杭州师范学院学报(自然科学版), 2001(6):21-24.
[11]沈波.数列与相应函数列的上、下极限间关系探讨[J].重庆师范大学学报(自然科学版), 2005(4):100-102.
[12]杨辉,宛金龙.数列上、下极限的注记[J].安庆师范学院学报(自然科学版), 2004(3):72-73.
[13]华东师范大学数学系.数学分析:上下册[M].3版.北京:高等教育出版社, 2008.
[14]陈纪修,於崇华,金路.数学分析:上下册[M].2版.北京:高等教育出版社, 2004.
[15]欧阳光中.数学分析:上下册[M].3版.北京:高等教育出版社, 2007.
[16]刘玉琏,傅沛仁.数学分析讲义:上下册[M].4版.北京:高等教育出版社, 2003.
[2]隋廷芳.上下极限的七个等价定义[J].内蒙古电大学刊, 1994(S4):4-7.
[3]王岩岩,刘伟.数列上下极限的研究[J].赤峰学院学报(自然科学版), 2011, 27(1):1-2.
[4]王振福,张建军.数列的上极限与下极限探析[J].包头职业技术学院学报, 2008(1):27-28.
[5]吐尼亚孜·库比.数列上、下极限的初等性质[J].新疆教育学院学报(汉文综合版), 1998, 14(1):54-55.
[6]艾斯卡尔·阿布力米提.数列上、下极限性质的推广应用[J].新疆教育学院学报, 2003(4):92-94.
[7]楼红卫.上下极限视角下的斯托尔茨公式和罗必塔法则[J].大学数学, 2017 33(1):96-99.
[8]程其襄,张奠宙,魏国强等编.实变函数与泛函数分析基础[M].3版.北京:高等教育出版社,2018.
[9]余国林,魏本成.关于上、下极限的一个新定理[J].大学数学, 2007(5):163-166.
[10]卢志康.上下极限的概念在极限教学中的作用[J].杭州师范学院学报(自然科学版), 2001(6):21-24.
[11]沈波.数列与相应函数列的上、下极限间关系探讨[J].重庆师范大学学报(自然科学版), 2005(4):100-102.
[12]杨辉,宛金龙.数列上、下极限的注记[J].安庆师范学院学报(自然科学版), 2004(3):72-73.
[13]华东师范大学数学系.数学分析:上下册[M].3版.北京:高等教育出版社, 2008.
[14]陈纪修,於崇华,金路.数学分析:上下册[M].2版.北京:高等教育出版社, 2004.
[15]欧阳光中.数学分析:上下册[M].3版.北京:高等教育出版社, 2007.
[16]刘玉琏,傅沛仁.数学分析讲义:上下册[M].4版.北京:高等教育出版社, 2003.