If you can show that the set of parameters obeying a property is non-empty, 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.
Point-set topology; partial differential equations
Problem: (Analytic continuation) Show that a real-analytic function that vanishes to infinite order at one point , is identically zero.
Solution: Let denote the assertion that vanishes to infinite order at . Then, by definition of real analyticity, if holds for some , then vanishes within the radius of convergence of the power series at , and so must then hold for all in a neighbourhood of . On the other hand, since all the derivatives of are continuous, if holds for a sequence converging to a limit , then also holds. Thus the set of where is true is non-empty, open, and closed, and is hence all of .
(Solving an ODE in a potential well)
This method can be viewed as a continuous analogue of mathematical induction (or conversely, induction can be viewed as a discrete analogue of the continuity method).