Real Analysis
实分析是数学分析的核心课程，以严格的公理化方法研究实数系上的函数理论。课程内容包括实数系的完备性、序列与级数的收敛理论、连续函数与一致连续性、微分学严格理论、黎曼积分理论、函数序列与函数项级数的一致收敛。学生将学习严格的数学证明方法，理解分析学的逻辑基础，培养严谨的数学思维能力。本课程是复分析、泛函分析、拓扑学等高级课程的先修基础。
Real Analysis is a core course in mathematical analysis, studying function theory on the real number system using rigorous axiomatic methods. Topics include completeness of real numbers, convergence of sequences and series, continuous functions and uniform continuity, rigorous differential calculus, Riemann integration, and uniform convergence of function sequences and series. Students will learn rigorous proof techniques and develop precise mathematical thinking.
深入研究魏尔斯特拉斯逼近定理，给出至少两种不同的证明方法（如伯恩斯坦多项式构造法、卷积磨光法）。探讨定理在不同函数空间的推广，研究逼近阶与函数光滑性的关系。结合具体例子进行数值验证，分析不同逼近方法的收敛速度与误差估计，展示分析学中构造性证明的思想方法。
Study the Weierstrass Approximation Theorem in depth, providing at least two different proofs (Bernstein polynomial construction, convolution mollification). Explore generalizations of the theorem to different function spaces, investigate the relationship between approximation order and function smoothness, and verify with concrete examples.