real-analysis-fundamentals skill

Overview

This skill is a rigorous analysis expert focused on limits, continuity, sequences, series, and the foundations of measure theory. It inspects proofs, clarifies definitions, and highlights logical gaps in arguments. Use it to raise the mathematical rigor of notes, lecture drafts, and problem solutions.

How this skill works

The skill parses mathematical statements and proofs, checks definitions against standard real analysis conventions, and identifies implicit assumptions or missing steps. It offers step-by-step corrections, alternative proofs when appropriate, and constructive examples or counterexamples. It also annotates where foundational measure-theoretic details are required and recommends authoritative sources for verification.

When to use it

• Validating the correctness and completeness of proofs in real analysis.
• Clarifying subtle definitions: convergence modes, measurability, null sets, sigma-algebras.
• Finding counterexamples or edge cases for proposed theorems or conjectures.
• Rewriting lecture notes or textbook drafts to improve rigor and pedagogical clarity.
• Preparing problem sets or solutions for undergrad/graduate real analysis courses.

Best practices

• Provide the full statement and context of the theorem or claim before asking for verification.
• Indicate which definitions and textbook conventions you follow (ε-δ, sequence vs net, Lebesgue vs measure).
• Request explicit proof steps you want checked rather than vague reviews.
• Use generated counterexamples as starting points; verify them with formal computation when needed.
• Mark corrected content as reviewed and update confidence metadata after independent source checks.

Example use cases

• Audit a proof that a sequence of functions converges almost everywhere but not in L1 and get a clear counterexample.
• Rewrite a textbook lemma about uniform continuity with explicit ε-δ constructions and edge-case notes.
• Identify and fill a missing measurability argument in a proof that uses Fubini/Tonelli.
• Produce a concise checklist for validating series convergence tests across different modes (pointwise, uniform, absolute).
• Transform intuitive lecture notes on limits into a rigorously stated theorem-proof pair suitable for handout distribution.

FAQ