Introduction to Analysis · Chapter 03

The real numbers and completeness

What 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.

sup A

∣ x - y ∣

Q \subset R

in f A

[a, b]

N \subset Z

n ⋂ I_{n}

2 \in / Q

Lessons

Build the foundation

01Number systemsEach number system was built to answer a question the one before it could not. Subtraction forces negative numbers, division forces fractions, and one further step is still needed. The fractions look complete, since between any two of them lies another, yet there are points on the line no fraction reaches.Read lesson →
02Order and absolute valueThe real numbers come with an order, and that order behaves well under addition but needs care under multiplication: multiplying an inequality by a negative number turns it around. Absolute value measures distance from zero, which turns questions about nearness into inequalities.Read lesson →
03IntervalsAn interval is a stretch of the line with no holes in it, and almost every set this course talks about is one. What decides an interval is not its length but whether it keeps its own endpoints, and that single distinction runs through the rest of the course: a closed interval has a largest element and an open one does not, a continuous function on a closed interval attains its maximum and on an open one may not.Read lesson →
04Upper and lower boundsA set is bounded above when some number is at least as large as everything in it. That number is called an upper bound, and it need not belong to the set. A set that has one upper bound has infinitely many, since anything larger works just as well.Read lesson →
05Supremum and infimum

Among all the upper bounds of a set, one may be smaller than every other. That least upper bound is called the supremum, and it is the sharpest ceiling the set admits. It sometimes belongs to the set and sometimes does not, which is exactly why it succeeds where the notion of a largest member fails.

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06CompletenessHere the question from the first lesson is answered. The rationals have gaps, and completeness says exactly what a gap is: a set that is bounded above but has no least upper bound. The real numbers are built so that this never happens, and that single guarantee is what the whole of analysis rests on.Read lesson →

07The Archimedean propertyCompleteness has been assumed; now it starts paying. The first thing it buys is a fact that looks too obvious to need a proof: no real number is larger than every natural number. It needs one because nothing said so far forbids a number sitting above the whole of the naturals — an ordered field can have such elements — and completeness is exactly what rules them out.Read lesson →

08Density of the rationals and the irrationalsBetween any two distinct real numbers, however close, there is a rational number — and there is an irrational one too. The rationals are therefore spread everywhere on the line, and so are the numbers that are not rational. Neither fact is about size: there are vastly more irrationals than rationals, and both kinds sit in every interval you can name.Read lesson →

09Nested intervalsA second face of completeness closes this chapter. Take a closed interval, then a closed interval inside it, then another inside that, forever. The nested interval property says something is left: at least one point belongs to every one of them. It is completeness in a form that can be used directly, and it is the engine of the bisection arguments that prove three of the largest theorems in this course.Read lesson →

10Chapter practiceThe chapter began with a question: the fractions look complete and are not. It ends with the answer stated precisely. This lesson mixes the ideas without saying which is which, because recognising which notion a question is about is most of the skill.Read lesson →

Mapping the lesson.

The real numbers and completeness

Build the foundation