I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero.
However, at the same time, they satisfy all the properties of real numbers - so for example, let's call such an infinitesimal $\epsilon$. Then $2 \epsilon$ and $3 \epsilon$ are greater than $\epsilon$. And most importantly, if they satisfy the rules known from real numbers, then they should also satisfy the archimedean property that states that it's always possible to give a number closer to zero than given number.
I've heard non-standard analysis simplifies some proofs. Erm, how can we prove theorems about real numbers in a system that includes objects that don't satisfy the properties of real numbers (infinitesimals in this case, because they are defined to be the number nearest to zero, which in fact doesn't exist)?