siddhartha-gadgil/leanaide/epsilon-delta

Skill: Epsilon-Delta Proofs

Pre-condition: Apply this skill when the goal is a limit, continuity, uniform continuity, convergence, or a metric/topological statement with explicit tolerances.

Goal: Choose the right tolerance parameter and verify the implication demanded by the definition.

Instructions:

1. Unfold the Definition: Write the exact quantified statement with epsilon, delta, neighborhoods, or sequence indices.
2. Fix the Tolerance: Let epsilon > 0 or the requested neighborhood be arbitrary.
3. Choose the Control Parameter: Define delta, N, or a smaller neighborhood in terms of the data.
4. Verify Positivity/Admissibility: Prove the chosen control parameter is valid.
5. Run the Estimate: Starting from the assumed closeness, derive the target closeness.
6. Use Standard Reductions: For sums, products, and compositions, split the tolerance into manageable pieces.

Common Failure Modes:

• Choosing delta after assuming a special x.
• Forgetting the 0 < |x-a| condition where needed.
• Not proving the chosen parameter is positive.