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For reference:

OK, first of all, I should come clean and admit that that entire "Topology before Calculus??" post was inspired by Hall's right on the money comments in the "Least Upper Bound Property" Thread. Least Upper Bound Property: https://www.physicsforums.com/showthread.php?t=207847

Topology before Calculus??? https://www.physicsforums.com/showthread.php?t=211744

L'Hospital's Rule: https://www.physicsforums.com/showthread.php?t=214203

The "topology before calculus" thread I worked on is abstract, nonessential, attempting to find intuitive idea of "connectedness" that can be taken as an axiom.. The "least upper bound" thread however is essential reading/information for proving calculus yourself, and the challenge problem is to prove the equivalent statements of the least upper bound property... but not necessarily the constructions such as dedekind cuts and completion of metric space, which are 4000 level..

Without the abstraction of the "topology before calculus" thread, you can take the definition of "connectedness" only of "open" sets, The axiom is then: R is connected:

an open set (in R) is connected if it is not a disjoint union of open sets.. by disjoint union could be disjoint with 2 sets, or a disjoint union of infinite number of open sets - either way it's disconnected.. I think in the special case of R you can define the number of holes in an open set U as the finite number of disjoint sets making up U minus 1.. and if it's an infinite union of disjoint open sets, it has an infinite number of holes.. But "holes" don't generalize to higher dimensions as connectedness/disconnectedness does, you need other notions for "hole" in general cases..

I created this current thread actually in reply to paniurelis in another thread.. promising to help him "prove calc for himself".. But given the amount of time I have spent on the two threads "Topology before Calc" and "L'Hospital" that no one will read, I don't intend to spend much time on this thread -- but hopefully others will help if questions are asked..

To make a long story short, I quote (butcher) a comment in Calculus on Manifolds by Spivak as I remember it (as wonk will point out - this is not an axiom for all of math):

That is, if you understand the various definitions of "limits", the epsilon-delta, and N-infinity definitions, then the theorems are pretty much trivial. My L'Hospital's post above is one of the more complex ones in differential calculus, and when you look at the summary in the thread I posted, the "proof idea" is too darn short..(still can be hard to remember) to be considered "difficult" -- assuming again you understand the limit definition/concept. You should not do hand waving like I did with the "≈" symbol if you don't understand how to translate it to epsilon-delta, N-Infinity. "the definitions are hard and the theorems are easy (trivial)".

You ultimately need to create a list of theorems in Calculus you think are interesting, and then proceed to prove them... the definitions of course involving epsilon-delta. As mentioned above, to do this you should be aware of the least upper bound property.

You should include compactness of [a,b] (look this one up and study it if it's the first time), equivalence of connectedness and least upper bound property, intermediate/extreme value theorems, mean value theorem, the fundamental theorem of calculus, and being able to prove the limit laws, differentiation laws, "integral laws", and the change of variable.

And that is my attempt on the direction to take to prove calculus for yourself..

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# How to prove calculus for yourself: summary

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