Quick description
If you can show that the set of parameters obeying a property is nonempty, open, and closed, and the parameter space is connected, then must be obeyed by all choices of the parameter. Thus, for instance, if one wants to prove a property for all in some interval , it suffices to establish the following three facts:

A base case for some ;

(Openness) If is true for some , then is true for all sufficiently close to ;

(Closedness) If is true for some sequence converging to a limit , then is also true.
Prerequisites
Pointset topology; partial differential equations
Example 1
(Solving an ODE in a potential well)