Your favorite Eureka moment for proofs

[ChinleShale]

For me, I think it was when I understood the δ-ε proofs in calculus. Somehow, I wasn't impressed with the Limits I'd learned shortly before. This was the first time I understood that this could be a really useful and different approach. "Arbitrarily close" wasn't a concept I'd known before.

These proofs are fundamental for understanding a lot of mathematics.

 

[ChinleShale]

To me there are at least three types of Eureka moments. One is when one understands a proof of a known theorem. A second is when one tries to prove a known theorem on one's own. The third is when one tries to discover a new theorem. This third one can take the form of seeing a relationship that unifies different examples. In math this Eureka moment often does not take the form of a rigorous proof but more of an insight. The rigorous proof comes later.

This last type is not unlike the process of discovering a new theory in any scientific field. @king vitamin 's post #32 describes this. In my own experience I was once studying a conjecture about the groups of isometries of a class of manifolds. I spent a year trying to construct a counter example. Reams of newsprint later, I saw a relationship that explained why no counter example exists. It took a lot of effort afterwards to find a rigorous proof.

 

[songoku]

My eureka moment happens every week here in PF homework section when I finally understand the hint given in post #2 of my thread after the helpers give another 50 hints