Analysis
What does it mean to approach without arriving?
Calculus works. It worked for a century and a half before anyone could say why, and the attempt to say why is this subject. Every definition here — limit, continuity, derivative, integral — exists because a plausible earlier version of it failed on a specific example. Those examples come first.
The Real Numbers
What is the line actually made of?
Two roads to the continuum
A square's diagonal is a length you can draw and not a ratio you can write, so the rationals have holes. The document builds the filling twice over and proves the two builds agree.
Why the line has no gaps
A bounded set with no smallest upper bound in sight. Once you have watched that happen, the least upper bound property stops being an assumption.
Sequences & Limits
How do you pin down a destination you never reach?
Approaching without arriving
Three plausible definitions of gets close to, each admitting a sequence that obviously should not qualify. The fourth attempt is epsilon, and every clause answers one failure.
Cauchy's trick: convergence without a limit
You can often tell a sequence is settling without knowing what it settles on — which is exactly what makes the reals constructible in the first place.
Continuity
What exactly does a gap cost you?
The function that cannot be drawn
Continuity is taught as drawing without lifting the pen, which is a picture rather than a definition — and there are functions for which the picture lies in both directions.
You cannot cross a line without touching it
The intermediate value theorem sounds like it needs no proof, which makes it the perfect place to show what a proof is for. What it needs is completeness.
Differentiation
Can a rate exist at a single instant?
The only line a curve will agree with
Average speed is unproblematic. Speed at an instant is a division by zero that mathematicians used anyway for two centuries because it kept giving right answers.
Somewhere, the average is achieved exactly
Between any two points on a smooth curve there is an instant where the instantaneous rate matches the average. The film makes it inevitable; the document is being written.
Integration
How much area is under a curve that has no formula?
Analysis, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume IV ↓