利用子列极限证明数列上下极限的性质

Proving Properties of Upper and Lower Limits of Sequences by Subsequence Limits

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.