Benquo: Even infinitesimal are not equal to zero. You don’t even need infinitesimals in differential calculus. Instead, you can think dx and dy are just variables. You let them approach zero to see what would happen at the limit, but you don’t set them equal to zero. I have always personally found infinitesimals a little disturbing, since one doesn’t really need them anywhere.
I am a little puzzled by this; I don’t know how they teach this stuff in the US, but in Finland, if my memory serves me correctly, they taught how this “proof” is wrong in elementary school. So only complete idiots would be fooled by this “logic”.
I think it was a bit later than elementary school, but showing why this “proof” is invalid was a test question at some point in my American academic career