Mathematics · Course 02

Introduction to Analysis

The real line done properly: completeness, limits, series, continuity, the derivative and the integral, each with the proof it rests on.

n \to \infty lim a_{n}

a \int b f (x) d x

f^{'} (x)

ε > 0

n = 1 \sum \infty a_{n}

sup A

f_{n} ⇉ f

R

∣ x - a ∣ < δ

k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k}

Course map

9 chapters

01Sets, functions and mathematical languageThe vocabulary the rest of the course is written in: membership, inclusion, the operations and their laws, families indexed by a set, and then functions — images, preimages, and what it means for two sets to have the same size.
02Logic and proofsStatements, connectives and quantifiers, then the four things you can do with them: prove directly, negate, argue indirectly, and climb the natural numbers one step at a time.
03The real numbers and completenessWhat the line has that the fractions do not. Order and distance, intervals, bounds, the least upper bound, the axiom that says there are no gaps, and the three theorems that come straight out of it.
04Sequences and limitsThe definition the rest of the course is built on, and how to write a proof with it: tolerance and place, what failure looks like, boundedness, uniqueness, the algebra, monotone convergence, subsequences, Cauchy sequences and the two limits a sequence always has.
05

Infinite series

Adding without end. Partial sums, the one thing every convergent series must do, the geometric family everything is compared against, the tests that follow from it, and why the order of the terms can decide the answer.

06Limits and continuity of functionsThe same idea as in the fourth chapter with the index replaced by a point: a tolerance answered by a distance. Then continuity, the two theorems that make a closed interval special, and the stronger condition a closed interval gives for free.

07DifferentiationThe derivative as a limit, and what one theorem about a flat point gives: the Mean Value Theorem, monotonicity, the counterexamples that show why each hypothesis is there, and Taylor with its remainder.

08The Riemann integralArea defined by squeezing: partitions, upper and lower sums, the criterion that decides integrability, which functions pass it, and the theorem that ties the integral to the derivative.

09Sequences and series of functionsWhat survives a limit taken one point at a time, and what needs the stronger kind: continuity, integration, differentiation, the test that gives uniformity for free, and power series.

Mapping the lesson.

Introduction to Analysis

9 chapters