Real Analysis

模块目标

Prove standard limit theorems from definitions, and identify when uniform convergence is required to pass limits inside integrals or derivatives.

课程大纲

1. 01 Sequences and limits 20h

   学习结果 Prove uniqueness of limits and the algebra of limits from the ε–N definition.

   形成性检查 书面推导

2. 02 Series 15h

   学习结果 Apply a convergence test and state its hypotheses; give a series where a test is inconclusive.

   形成性检查 反例

3. 03 Continuity 20h

   学习结果 Prove that a uniform limit of continuous functions is continuous, or exhibit a pointwise failure.

   形成性检查 书面推导

4. 04 Differentiation 20h

   学习结果 Prove the mean-value theorem from Rolle, listing hypotheses.

   形成性检查 书面推导

5. 05 Riemann integration 25h

   学习结果 Define the Riemann integral via upper and lower sums and prove integrability of a continuous function on a compact interval.

   形成性检查 书面推导

6. 06 Uniform convergence 20h

   学习结果 Give a counterexample where pointwise convergence does not preserve the integral.

   形成性检查 反例

7. 07 Metric language (introduction) 15h

   学习结果 Translate sequential compactness on ℝ into a metric statement used later.

   形成性检查 四选一检查

8. 08 Synthesis 15h

   学习结果 Produce one proof that combines a definition, a theorem, and a sharp counterexample.

   形成性检查 书面推导

学习结果

Write a complete ε-style proof for a named theorem, and give a counterexample where a hypothesis is dropped.

学科技能

Sequences, series, continuity, differentiation, Riemann sums, uniform convergence.

可迁移技能

Keep hypotheses visible; replace informal “close” language with a quantifier pattern.