This reminded me of a talk by Mircea Mustata in which he mentioned that non-standard analysis type arguments were used to prove some things related to algebraic geometry. I can't remember what the talk was about, but I found the paper that it was based on: de Fernex and Mustata - Limits of log canonical thresholds.
The paper mentions that later Kollár found proofs avoiding these techniques (but they are similar in spirit).
