If we wanted to prove certain statements for every element of a set $S$, a possible approach is to prove the statement for a certain dense subset $S'\subset S$ (with respect to a certain metric), then show that if the statement is true for $S'$ then it is for $S$. Sometimes, this approach can reduce the proof of a difficult theorem to something nearly trivial, such as the following proof of Cayley-Hamilton:
Theorem. Let $A\in M_n(\mathbb C)$, and let $p(\lambda)$ be the characteristic polynomial of $A$. Then $p(A)=0$.
Proof. The statement is trivial for diagonal matrices. If $A$ is diagonalisable, write $A=P\Lambda P^{-1}$ for the diagonal matrix of eigenvalues $\Lambda$. Since the characteristic polynomial of $A$ is the same as that of $\Lambda$, we have $p(A)=Pp(\Lambda)P^{-1}$, and since $p(\Lambda)=0$, we have $p(A)=0$. By the density of diagonalisable matrices in $M_n(\mathbb C)$, we recover the general statement for all matrices in $M_n(\mathbb C)$.
What are some more interesting instances of this method to prove a difficult theorem?