siddhartha-gadgil/leanaide/density-approximation

Skill: Density and Approximation

Pre-condition: Apply this skill when a statement should be proved first on a dense or simpler class and then extended by continuity, closure, boundedness, or limiting control.

Goal: Reduce a hard object to approximating simple objects and pass the result through a limiting argument.

Instructions:

1. Choose the Dense/Simple Class: Identify polynomials, simple functions, smooth functions, finitely supported objects, rational points, compactly supported objects, or another tractable subset.
2. Prove the Result on the Simple Class: Use direct computation or known theorems where the structure is easier.
3. Approximate the General Object: Produce a sequence/net or error bound that converges to the target object.
4. Control the Error: Use continuity, uniform bounds, dominated convergence, closure, completeness, or stability estimates.
5. Pass to the Limit: State exactly which topology or norm is used and conclude the general case.

Common Failure Modes:

• Picking a dense subset in the wrong topology.
• Lacking uniform estimates needed to pass a nonlinear operation to the limit.
• Proving convergence but not preservation of the target property.