> The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason?

I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus. Compactness is a property of topological spaces that, to an approximation, is a generalization of sets being finite or infinite. For example, with the discrete topology, a set is compact iff it is finite. There are many related notions of compactness. In R^n, a set is compact (and sequentially compact) iff it is closed and bounded.

Compactness is important for some ideas related to calculus, but it's not related to formulating calculus. For example, if a continuous function maps from a compact space to R, then it achieves a maximum/minimum (this can be seen of a generalization that there is always a maximum/minimum of a finite set of real numbers, but not necessarily for an infinite set).

The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number. This is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus.

The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits; that's what's taught in both calculus classes (at least to an extent) and analysis classes. The weird thing is that calculus is stuck with Leibniz's notation, which does, in a sense, refer to infinitesimals. I think that's what you're really thinking of (rather than compactness) as how you can formulate calculus without infinitesimals. The thing is that, save notation, this is how calculus is taught today.