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 This approach focuses on infinitesimals, also known as ultrasmall numbers.  Mathematics with ultrasmall numbers can be practiced in a style that is just as informal and natural as traditional treatments but has important advantages, such as simpler proofs and the ability to prove results without needing to master notions of supremum or compactness Analysis has been taught using ultrasmall numbers for more than ten years in some high schools in Geneva (Switzerland). We give as an example of what things look like: The definition of continuity. $$f$$  is continuous at $$a$$ if $x\simeq a\Rightarrow f(x)\simeq f(a)$   Where the $$\simeq$$ symbol reads "ultraclose to".