REFERENCE SHEET

Completeness & sequences

Completeness guarantees that approximations whose internal error vanishes have a limit in the space.

Before this Limits & continuity, Proof techniques

∀ε>0 ∃N ∀m,n≥N: |xₘ−xₙ|<ε (Cauchy)

Definitions

Supremum

The least upper bound of a set. It need not belong to the set.

Cauchy sequence

A sequence whose sufficiently late terms are arbitrarily close to one another, as in the display.

Complete metric space

A metric space in which every Cauchy sequence converges to a point of that space.

Key theorems

Completeness of ℝ

Every nonempty subset of ℝ bounded above has a real supremum. Equivalently, every real Cauchy sequence converges in ℝ.

Monotone convergence for sequences

Every bounded monotone real sequence converges. An increasing bounded sequence converges to the supremum of its values.

Bolzano–Weierstrass

Every bounded sequence in ℝⁿ has a convergent subsequence.

Notable applications

Existence proofs

Define a candidate solution as a limit and use completeness to ensure it exists.

Iterative computation

Proving an iteration is Cauchy can establish convergence without already knowing its limit.

Work it out

Try before revealing

01 / Compute

Find sup{1−1/n:n≥1}. Is it attained?

Hint

Every element is below 1 but can get arbitrarily close.

Solution

The supremum is 1. It is not attained because 1−1/n<1 for every positive integer n.

02 / Prove

Why is every convergent real sequence Cauchy?

Hint

Insert the limit between two late terms.