My instructor for Epsilon Delta proofs and limits would always talk about "his cousin in Romania" picking the Epsilon and him picking the Delta.
i.e. forall epsilon > 0. exists delta > 0. forall d with |d| < delta. |f(x) - f(x+d)| < epsilon.
If we had a proof, no matter what epsilon his cousin from Romania picked, we could always find a new delta which would satify his cousin and let him pick the worst d in range.
This worked better than just saying "pick any epsilon", as it convayed the adversarial approach better.
Another book I read used the Devil as the one you are trying to convince, but it's nowhere near as fun as "his cousin from Romania".
i.e. forall epsilon > 0. exists delta > 0. forall d with |d| < delta. |f(x) - f(x+d)| < epsilon.
If we had a proof, no matter what epsilon his cousin from Romania picked, we could always find a new delta which would satify his cousin and let him pick the worst d in range.
This worked better than just saying "pick any epsilon", as it convayed the adversarial approach better.
Another book I read used the Devil as the one you are trying to convince, but it's nowhere near as fun as "his cousin from Romania".