Real Analysis

Module aims

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

Course outline

1. 01 Sequences and limits 20h

   Learning result Prove uniqueness of limits and the algebra of limits from the ε–N definition.

   Formative check Written derivation

2. 02 Series 15h

   Learning result Apply a convergence test and state its hypotheses; give a series where a test is inconclusive.

   Formative check Counterexample

3. 03 Continuity 20h

   Learning result Prove that a uniform limit of continuous functions is continuous, or exhibit a pointwise failure.

   Formative check Written derivation

4. 04 Differentiation 20h

   Learning result Prove the mean-value theorem from Rolle, listing hypotheses.

   Formative check Written derivation

5. 05 Riemann integration 25h

   Learning result Define the Riemann integral via upper and lower sums and prove integrability of a continuous function on a compact interval.

   Formative check Written derivation

6. 06 Uniform convergence 20h

   Learning result Give a counterexample where pointwise convergence does not preserve the integral.

   Formative check Counterexample

7. 07 Metric language (introduction) 15h

   Learning result Translate sequential compactness on ℝ into a metric statement used later.

   Formative check Four-option check

8. 08 Synthesis 15h

   Learning result Produce one proof that combines a definition, a theorem, and a sharp counterexample.

   Formative check Written derivation

Learning results

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

Subject skills

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

Transferable skills

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