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A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
The least upper bound property is the essential property of real numbers that permits the main theorems of calculus. It is the reason we use this large set, rather than, say, the algebraic numbers. It uniquely characterizes the real numbers as an extension of the rational numbers - see Theorem \(8.23\) for a precise statement.
5 days ago · Let S be a nonempty set of real numbers that has an upper bound. Then a number c is called the least upper bound (or the supremum, denoted supS) for S iff it satisfies the following properties: 1. c>=x for all x in S. 2. For all real numbers k, if k is an upper bound for S, then k>=c.
The least upper bound property indicates that every non-empty set of real numbers that is bounded above has a least upper bound within the reals. In contrast, some sets of rational numbers can be bounded above but do not have a least upper bound within the rationals.
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Let S⊂RS⊂R be a non-empty subset of the set of real numbers such that SS is bounded above. Then SS admits a supremum in RR. This is known as the least upper bound property of the real numbers.
Suppose that S⊆R≥0S⊆R≥0 has the positive real number UU as an upper bound. Then R≥0R≥0 can be represented as a straight line LL whose sole endpoint is the point OO. Let l0∈R≥0l0∈R≥0 be the standard unit of length. There exists a unique point X∈LX∈L such that U⋅l0=OXU⋅l0=OX. Furthermore, if x∈Sx∈S, then: 1. f(x)=x⋅l0f(x)=x⋅l0 where ⋅⋅ denotes (real)...
Let S be bounded above. Let L be the set of real numbersdefined as: 1. α∈L⟺∃x∈S:α
The least upper bound property of Ris also known as: 1. the supremum principle 2. the completeness property 3. the continuum property (although this is also used to encompass the Greatest Lower Bound Property, a complementary result).
1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter 1: Fourier Series: 1.1 Basic Concepts: 1.1.2Definitions1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): 1: Review of some real analysis: §1.1: Real Numbers: Axiom 1.1.41977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): §2: Continuum Property: §2.4: The Continuum Property1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order properties (of real numbers): (5): Completeness propertyTheorem (Least Upper Bound Property): Every non-empty subset of $R$ that is bounded above has a least upper bound. The easy proof for Dedekind cuts shows that the least upper bound of a non-empty, bounded-above set $X$ of Dedekind cuts is obtained by taking the union of all the Dedekind cuts in $X$.
Condition (i) states that \(q\) is an upper bound of \(A,\) while (ii) implies that no smaller element \(p\) is such a bound (since it is exceeded by some \(x\) in A). When combined, (i) and (ii) state that \(q\) is the least upper bound.
