%0 Journal Article
%T 柯西收敛原理及其应用研究<br>Research on the Cauchy Convergence Criterion and Its Applications
%A 贾瑞玲
%A 孙铭娟
%A 张冬燕
%J Pure Mathematics
%P 153-160
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.155164
%X 柯西收敛原理是实数完备性理论的核心工具&#65292;它通过评估序列项之间的自洽性判定收敛性&#65292;摆脱了对极限值的依赖&#65292;兼具理论深度与应用广度。然而&#65292;由于其抽象性与严格性使其成为学生学习过程中的难点&#65292;深入剖析柯西收敛原理的结构特征与应用场景&#65292;不仅有助于学生更好地理解数学分析的基本概念&#65292;而且对培养他们的数学思维和分析能力具有重要意义。<br>The Cauchy Convergence Criterion is a core tool in the theory of real number completeness. It determines convergence of a sequence by evaluating the self-consistency among its terms, thus avoiding reliance on limit values. It combines theoretical depth with broad application. However, due to its abstractness and strictness, it poses a challenge for students during their learning process. A thorough analysis of the structural features and application scenarios of Cauchy Convergence Criterion not only helps students better understand the fundamental concepts of mathematical analysis, but also plays a significant role in cultivating their mathematical thinking and analytical abilities.
%K 柯西收敛原理&#65292
%K 实数完备性&#65292
%K 一致收敛性&#65292
%K 自洽性&#65292
%K 数学分析<br>Cauchy Convergence Criterion
%K Completeness of Real Numbers
%K Uniform Convergence
%K Self-Consistency
%K Mathematical Analysis
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=114994