At project-perelman, We have inevitably started delving into Cauchy sequences and real analysis because if you really want to believe calculus proofs and have enough genuine interest and analytical mind, you will discover very fast that there are missing foundations and hand-waving proofs all over the place if you do not go deeper. This is best expressed by Karl Hahn in his excellent website.
As the first thing in introducing calculus he says:
Calculus deals with properties of the real numbers. In order to understand calculus you must first understand what it is about the real numbers that separates them from other kinds of numbers we use from day to day."
As an example, my (failed) proof (http://jadnohra.net/release/math/evt.pdf) of the Extreme Value Theorem does not perfectly work because I could not prove that my induction would cover the whole space of inputs. This can only be done by digging deeper than calculus.
Discussing with Tom we agreed that this is not explained in the majority of cases because there is no real interest in Calculus, and we have many times discussed how it ends up being used as a way to mechanically solve meaningless (to the student) exercises, and that does not stop at school. In any case, for the interested, it is inevitable.
"There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple"
It puts in words what we many times feel but cannot express, when something mathematical feels 'natural'. Worth memorizing!
To all of this, I will add this unrelated and brilliant quote that my lovely wife just sent me: