[MisterMashable]

The most fruitful definition of a real number is as a limit of a Cauchy sequence. That way is much more useful in proving theorems.

Using infinitesimals is logically valid (alternative real analysis), useful for physics and other practical calculations but not at all helpful proving theorems.

Might I add that the concept of 'nearness' introduced by Riesz is the contrapositive of the usual limit definition and might be the way real analysis is taught 100 years from now.

Hyperreals are much more involved than mere epsilontics as they include all kinds of infinities. It's so mind blowing that I simply must defer to minds like Conway to play with such things.