实分析 RA-001

Sequences and Convergence in the Real Numbers

Definitions and basic properties of convergent sequences, Cauchy sequences, and the completeness of the reals.

Goals

We introduce the notion of convergence for real sequences, establish key properties, and state the completeness axiom.

Notation
  • or denotes a sequence of real numbers.
  • is the set of positive integers.

Convergence

Definition: Convergence of a Sequence

A sequence of real numbers converges to a limit if for every there exists such that

We write , or simply .

Example

The sequence converges to . Given , choose (by the Archimedean property). Then for :

Theorem: Uniqueness of Limits

If and , then .

Proof

Suppose . Set . There exist such that for and for . For :

This is a contradiction, so .

Cauchy Sequences

Definition: Cauchy Sequence

A sequence is Cauchy if for every there exists such that

Theorem: Completeness of the Reals

A sequence of real numbers converges if and only if it is Cauchy.

Remark

Completeness is what distinguishes from . The sequence of rational approximations to is Cauchy in but does not converge in . The real numbers are, by construction, the completion of the rationals.

Further Reading

  • Rudin, W. Principles of Mathematical Analysis, Chapter 3 (1976)
  • Tao, T. Analysis I, Chapter 6 (2016)