Skip to main content Accessibility help
×
×
Home
  • Print publication year: 2012
  • Online publication date: August 2012

Chapter 2 - The real numbers

Summary

The complete ordered field of real numbers

The real numbers form an ordered field ℝ containing the rationals with an additional property called completeness that the rationals do not satisfy. We need some preliminary definitions to be able to say what completeness means.

2.1. Definition. An upper bound for a subset A ⊂ ℝ is an element b ∈ ℝ such that ab for all aA. If A has an upper bound, then A is said to be bounded above.

A lower bound for a subset A ⊂ ℝ is an element b ∈ ℝ such that ba for all aA. If A has a lower bound, then A is said to be bounded below.

If A is bounded above and bounded below, then A is said to be bounded.

2.2. Example. Consider the interval [0, 1] = {x ∈ ℝ : 0 ≤ x ≤ 1}. It is bounded above, for example by the upper bound 1. The upper bounds for [0, 1] are precisely the numbers b with b ≥ 1. Thus 1 is the smallest upper bound for [0, 1], and it is of course also the largest element of [0, 1].

Now consider the interval (0, 1) = {x ∈ ℝ :0 < x < 1}, also bounded above, for example by 1. It has the same upper bounds as [0, 1]. Namely, if b ≥ 1 and x ∈ (0, 1), then x < 1 ≤ b, so b is an upper bound for (0, 1).

Recommend this book

Email your librarian or administrator to recommend adding this book to your organisation's collection.

Lectures on Real Analysis
  • Online ISBN: 9781139208604
  • Book DOI: https://doi.org/10.1017/CBO9781139208604
Please enter your name
Please enter a valid email address
Who would you like to send this to *
×