Gromov's proof that finitely generated groups with polynomial growth are virually nilpotent. The ingenius step is to consider a scaling limit of the usual metric on the Cayley graph of the finitely generated group.

Of course the details are messy and to get the final conclusion one has to rely on a lot of deep results on the structure of topological groups. However, already the initial idea is breathtaking.

Gromov's proof that finitely generated groups with polynomial growth are virtually nilpotent. The ingenious step is to consider a scaling limit of the usual metric on the Cayley graph of the finitely generated group.

Of course the details are messy and to get the final conclusion one has to rely on a lot of deep results on the structure of topological groups. However, already the initial idea is breathtaking.