证明了收敛数列的一个简单性质。
Prove one simple nature of convergence for sequence of Numbers, Popularize this to convergence series.
最后讨论了一致收敛函数列与函数项级的性质。
Finally, out of uniform convergence of function and function of the nature of class.
定理1（极限的唯一性）如果数列收敛，那么它的极限唯一。
Theorem 1 (Uniqueness of Limit) If the sequence is convergent, then its limit is unique.
通过巧妙地构造辅助数列，应用致密性定理、柯西收敛准则来证明闭区间上连续函数的介值性定理。
We proved the intermediate value theorem for continuous function at closed interval by constructing auxiliary sequence ingeniously and applying compact theorem as well as Cauchy convergence criterion.
主要把数列收敛的一些性质引进到随机变量依概率收敛中来，并加以证明。
The property in convergence of sequence of number was imported to convergence in probability and proved it in this paper.
第二部分是在一致收敛条件下函数列、函数项级数以及含参量反常积分的性质。
The second part is in uniform convergence conditions function series, function and parameter improper integral. We properties.
为了得到关于弱集值渐近鞅的收敛性质，首先证明了支撑函数列的极限亦为一支撑函数。
In order to get the convergence properties of the weak set-valued Amart, we firstly proved the theorem that the limit of support functions is a support function.
连续、一致连续、一致收敛和等度连续是函数或函数列非常重要的性质。
Continuity, uniform continuity, uniform convergence and equicontinuity are very important qualities of functions or sequence of functions.
对一个数列的奇子列和偶子列收敛且极限相等则原数列收敛的性质做了推广。
The property of the convergent sequence about even subsequence and odd subsequence is generalized.
利用分析方法讨论随机选择函数列的收敛性问题，得到了一些有关的强大数定律。
By using the analytic approach, the problems of the convergence about the random selection function sequence are discussed and some strong laws of large Numbers are obtained.
应用单调有界定理证明一类数列的收敛过程中，一般高等数学和数学分析教材中，处理的思路方法不易想到或过程较为繁琐。
In many current textbook, the monotone bounded theorem is used to the proof the convergence of a kind of but this method is uneasy to series, understand.
这些特性能更好地描述模糊值可测函数列和K -拟可加模糊数值积分序列的收敛性。
These characteristics can be used well to describe the convergence properties of sequences of fuzzy valued measurable functions and K-quasi-additive fuzzy number valued integrals.
此文对“单调有界数列必收敛”两个条件单调，有界的证明方法加以归纳，并就两个条件的关系及一类特殊情况加以讨论，得出结论。
The essay is a summary concerning how to demonstrate the two prerequisites monotone and bounds in "monotonous and boundary number line necessarily converge."
定理7.1。如果一个数列单调并且有界，这个数列才能收敛。
THEOREM 7.1. A monotonic sequence converges if and only if it is bounded.
我们乐意处理单调递增数列或单调递减数列，因为特别容易确定数列的收敛或发散。
Monotonic sequences are pleasant to work with because their convergence or divergence is particularly easy to determine .
文章通过对无穷小量与无穷大量的阶的概念研究，用阶的估计讨论数学分析中数列、函数及级数收敛问题，也为收敛问题深入研究提供了一种方法。
This paper studies the concept of infinitesimal and infinity, and discusses the convergence of sequence, function and series with the estimation of the orders.
文章通过对无穷小量与无穷大量的阶的概念研究，用阶的估计讨论数学分析中数列、函数及级数收敛问题，也为收敛问题深入研究提供了一种方法。
This paper studies the concept of infinitesimal and infinity, and discusses the convergence of sequence, function and series with the estimation of the orders.
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