Calculus using proximities: a mathematical approach in which students can actually prove theorems

What is it about?

This article attempts at showing in the least technical way how a concept of ultrasmall numbers can lead to more intuive proofs and be useful at introductory level.

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Why is it important?

Flavours of nonstandard analysis have already made attemps at using infiniteismals for introducing analysis. All these approaches had technica difficulties which in a certain way defeated the purpose of the high school teacher. We believe this is the first time a mathematically correct theroy with infinitesimal type objects is really available for introducing the subject.