I recently read in a book about a proof that Archimedes did. I don't remember the exact details and I don't have the book on me right now, but it involved proving an equality. So let's say proving that $A=B$. In order to do this, he first assumed that $A>B$ and found a contradiction. He then assumed that $A<B$ and once again, found a contradiction. Therefore, he concluded, it must be that $A=B$.
I had never thought of this proof method or encountered it, so I thought it was very neat. Of course you're doing twice as much proving, if you like, but I feel like proving an inequality can often be easier than an equality because you can 'neglect' certain terms which don't effect the inequality.
I don't know if it is just the education I have had, but I am an undergraduate now and, like I said, I've never encountered this proof. I feel this is a shame because it would be a handy tool to know for any student. So my question is, why is this type of proof not used more often and taught?
The book was Journey through genius by William Dunham