Introduction to Analysis · Chapter 04

Sequences and limits

The 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.

a_{n} ⟶ L

∣ a_{n} - L ∣ < ε

n \to \infty lim a_{n}

ε > 0

n \geq N

a_{n_{k}}

lim sup a_{n}

lim inf a_{n}

Lessons

Build the foundation

Convergent sequences are boundedA sequence that settles cannot run away, and the proof has a shape worth learning because it is used again and again: the definition controls the tail, a finite list is handled by taking a maximum, and the two are put together. The converse is false, and the sequence that shows it is the one from the last lesson — bounded, and settling nowhere.

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06Uniqueness of the limitA sequence cannot settle towards two different numbers. The reason is geometric: two different numbers have a gap between them, and if the terms were eventually within half that gap of each one, they would have to be in two places at once. The proof turns that picture into three lines, and it is the first place in the course where the tolerance is chosen to fit the problem rather than taken as given.Read lesson →

07The algebra and order of limitsLimits survive arithmetic. If two sequences settle, so do their sum, their difference and their product, and each settles where you would hope. Division needs one extra care: the sequence you divide by must not settle at zero. These rules are what let a complicated limit be broken into pieces.Read lesson →

08Monotone convergenceHere completeness earns its keep. A sequence that only rises, and never passes some ceiling, has to settle somewhere, and the place it settles is the least such ceiling. Nothing about the terms tells you what that number is; the supremum is what produces it. This is the first theorem of the course that could not be proved without the completeness axiom.Read lesson →

09Subsequences and Bolzano–WeierstrassA subsequence is what is left when terms are struck out, provided infinitely many survive and their order is kept. Two facts make the idea powerful. A convergent sequence drags every subsequence to the same limit, so two subsequences with different limits prove a sequence diverges. And every bounded sequence, however badly it behaves, has at least one subsequence that does converge — which is the deepest theorem of the chapter and the one the rest of the course leans on.Read lesson →

10Cauchy sequences and the Cauchy criterionEvery test for convergence so far has needed the limit in advance. That is a real limitation: most sequences that arise are defined by a rule, and their limits have no name. A Cauchy sequence is one whose terms eventually crowd together — a condition that mentions no limit at all — and the criterion of this lesson says that in the real numbers, crowding together is the same as converging. It is completeness wearing its most usable face.Read lesson →

11The upper and lower limitsA bounded sequence need not have a limit, but it always has two numbers that behave almost as well. Look at the tail from the nth place on and take its supremum; as n grows that supremum can only fall, and a falling sequence bounded below settles. What it settles at is the upper limit, and the mirror construction gives the lower one. They always exist, they trap the sequence, and they are equal exactly when the sequence converges.Read lesson →

12Chapter practiceThis chapter turned the whole course into a single definition and then took it apart. The questions here arrive without saying which idea they test, which is the state in which mathematics is normally met.Read lesson →

Mapping the lesson.

Sequences and limits

Build the foundation