Course review · Introduction to Analysis

Intuition

One idea runs through the whole course: a tolerance is named and something must answer it. In chapter four the answer was a place along the index, in chapter six a radius about a point, in chapter eight a partition, and in chapter nine a stage of a sequence of functions. One axiom is behind all of them, and this lesson is where the shape of the whole thing is worth looking at.

A single tool used on nine different materials. Once the grip is learnt, the differences between the chapters are differences in what is being cut, not in how the tool is held.

What the course rests on

Completeness is the only axiom. From it come the Archimedean property, density, nested intervals, monotone convergence, Bolzano–Weierstrass, the Cauchy criterion, the Intermediate and Extreme Value theorems, Heine–Cantor, the integrability of continuous functions and the Fundamental Theorem. Every one of those is spent somewhere later, and the course is arranged so that nothing is used before it is proved.

S\ne\varnothing\ \text{bounded above}\implies\sup S\ \text{exists}

The facts this course takes on trust

That every real number has a decimal expansion, used once in the diagonal argument.
That the exponential, logarithm and trigonometric functions exist and are continuous, used only for examples and never in a proof.
Riemann's rearrangement theorem in full: the non-negative case is proved and the conditional case is stated.
That a continuous function exists that is differentiable nowhere, and that its standard construction is a series; the series is not summed here.
That the Riemann sum definition of the integral gives the same class of integrable functions as the upper and lower sums.

Practice

One Axiom, Nine Chapters

Everything that distinguishes the reals from the rationals in this course comes from one statement.

S\ne\varnothing\ \text{bounded above}\implies\sup S\ \text{exists}

Try it

Which statement is the axiom this course assumes, rather than a theorem proved from it?