Here's the whole file of Fundamentals of Analysis and more.
I cut it up in pieces below.
Check out the video's and playlists made during the first corona lock down.
In the pdf-files below the reference links don't work globally, in the big pdf they do.
About.... (written some 30 months after 10-02-23).
Inhoudsopgave (contents).
Learning goals Chapter 1-11 for a first year Mathematical Analysis course.
Victor was in the audience when I started the course on February 6 in 2019.
Early this millennium I taught a similar course from The Way of Analysis book of Strickartz. I kept the
pre-canvas-blackboard web environment of that course with all the
blackboards (in Dutch). Strickartz did everything without epsilons.
In the introductionary Chapter 1
I begin with YBC7289 and the square root of 2.
Then I let Archimedes bring in the epsilons for the reciprocal of 3 in a
pyramidal
context of sums of consecutive squares. Decimal expansions are used to introduce the geometric series, basically replacing 10 by n.
Compound interest introduces Euler's number.
In the context of Heron's method for the square root of 2 Chapter 2 is first about limits of bounded monotone sequences of real numbers. Then I introduce the more general concept of convergence. Cliff hanger: how to get rid of the limit being part of the convergence definition?
Chapter 3 uses Han Peters' elegant observation over beer one day that every sequence of numbers on the number line can be turned into a monotone sequence by deleting (not too) many of its numbers. Not by man though as you may infer from the proof I borrowed from Thomas Rot.
Han's observation resolves the cliffhanger via Cauchy's definition (or criterion if you like) for convergence of a sequence of real numbers.
The other highlight in Chapter 3 is the Banach Contraction Theorem in the real number context, which requires the concept of closedness for sets consisting of real numbers.
There's also a section about the real numbers but only after the exercise section and then one about (absolute and unconditional) convergence of series. Finally there's a teaser is about difference quotients.
Chapter 4 is
about real valued continuous functions defined on the closer interval [a,b].
Here a and b are fixed real numbers with a smaller than b.
Chapter 5
is about C([a,b]),
the Banach algebra of real valued continuous functions defined on the closer interval [a,b], as the first example of a complete metric space.
The story so far.
Chapter 6.
Chapter 7.
Chapter 8.
Chapter 9.
John Landen's Residual Analysis (1764) was yet not known to me
when I opted for the algebraic approach to differentiation.
Ik lees nu pas Struik.
Chapter 10.
Chapter 11.
Chapter 12.
Chapter 13.
Chapter 14.
Chapter 15.
Chapter 16.
Chapter 17.
Chapter 18.
Chapter 19.
Chapter 20.
Chapter 21.
Chapter 22.
Chapter 23.
Chapter 24.
Chapter 25.
Chapter 26.
Chapter 27.
Chapter 28.
Chapter 29.
Chapter 30.
Chapter 31.
Chapter 32.
Chapter 33.
Chapter 34.
Chapter 35.
Chapter 36.
Chapter 37.
Chapter 38.
Chapter 39.
Chapter 40.
Chapter 41.
Chapter 42.
Chapter 43.
Chapter 44.
Chapter 45.
Chapter 46.
Chapter 47.
Chapter 48.
Chapter 49.
Chapter 50.
Chapter 51.
Chapter 52.
Ik lees nu pas Struik.
Bijvoorbeeld dit.