柯西收敛原理及其应用研究
Research on the Cauchy Convergence Criterion and Its Applications

柯西收敛原理是实数完备性理论的核心工具，它通过评估序列项之间的自洽性判定收敛性，摆脱了对极限值的依赖，兼具理论深度与应用广度。然而，由于其抽象性与严格性使其成为学生学习过程中的难点，深入剖析柯西收敛原理的结构特征与应用场景，不仅有助于学生更好地理解数学分析的基本概念，而且对培养他们的数学思维和分析能力具有重要意义。
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.