How To Find The Least Upper Bound

Every non-empty subset ${\displaystyle Chiliad}$ of the real numbers ${\displaystyle \mathbb {R} }$ which is bounded from above has a to the lowest degree upper bound.

In mathematics, the least-upper-leap property (sometimes chosen completeness or supremum property or l.u.b. property)^[1] is a fundamental holding of the real numbers. More mostly, a partially ordered set X has the least-upper-leap property if every not-empty subset of X with an upper spring has a least upper jump (supremum) in 10 . Not every (partially) ordered set has the least upper bound property. For example, the set ${\displaystyle \mathbb {Q} }$ of all rational numbers with its natural order does non have the least upper bound property.

The least-upper-leap belongings is one form of the completeness precept for the real numbers, and is sometimes referred to every bit Dedekind completeness.^[2] It can be used to testify many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the farthermost value theorem, and the Heine–Borel theorem. It is usually taken every bit an axiom in constructed constructions of the real numbers, and it is also intimately related to the construction of the real numbers using Dedekind cuts.

In order theory, this holding can be generalized to a notion of completeness for whatever partially ordered gear up. A linearly ordered set that is dense and has the least upper jump property is called a linear continuum.

Statement of the holding [edit]

Statement for existent numbers [edit]

Let Due south be a non-empty set of real numbers.

• A real number 10 is called an upper spring for Due south if 10 ≥ due south for all s ∈ Southward .
• A real number 10 is the to the lowest degree upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of South .

The least-upper-jump property states that whatever non-empty set of existent numbers that has an upper bound must have a to the lowest degree upper leap in real numbers.

Generalization to ordered sets [edit]

Red: the gear up ${\displaystyle \left\{x\in \mathbf {Q} :x^{2}\leq ii\right\}}$ . Blueish: the set up of its upper bounds in ${\displaystyle \mathbf {Q} }$ .

More than more often than not, i may ascertain upper bound and to the lowest degree upper jump for any subset of a partially ordered set Ten , with "existent number" replaced by "element of 10 ". In this case, we say that X has the least-upper-bound belongings if every non-empty subset of X with an upper bound has a to the lowest degree upper bound in X .

For example, the set Q of rational numbers does not have the to the lowest degree-upper-spring property under the usual gild. For case, the ready

${\displaystyle \left\{x\in \mathbf {Q} :ten^{2}\leq 2\right\}=\mathbf {Q} \cap \left(-{\sqrt {2}},{\sqrt {ii}}\right)}$

has an upper bound in Q , merely does not accept a least upper leap in Q (since the foursquare root of 2 is irrational). The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers equally the least upper premises of certain subsets of the rationals.

Proof [edit]

Logical condition [edit]

The least-upper-bound property is equivalent to other forms of the completeness precept, such every bit the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the structure of the real numbers used: in the synthetic approach, the property is usually taken as an precept for the real numbers (see least upper spring axiom); in a constructive approach, the belongings must exist proved equally a theorem, either directly from the construction or equally a event of some other form of completeness.

Proof using Cauchy sequences [edit]

It is possible to evidence the least-upper-bound property using the assumption that every Cauchy sequence of existent numbers converges. Permit S exist a nonempty ready of real numbers. If S has exactly ane chemical element, and then its only element is a least upper bound. So consider Southward with more than than one element, and suppose that S has an upper spring B _i . Since S is nonempty and has more than one element, there exists a real number A ₁ that is not an upper jump for South . Define sequences A ₁, A _two, A ₃, ... and B ₁, B ₂, B ₃, ... recursively as follows:

1. Cheque whether (A_n + B_n ) ⁄ two is an upper bound for S .
2. If information technology is, let A _n+i = A_north and let B _{northward+ane} = (A_n + B_north ) ⁄ ii.
3. Otherwise there must exist an element s in South so that due south>(A_n + B_n ) ⁄ 2. Let A _north+1 = s and let B _n+1 = B_{due north} .

Then A ₁ ≤ A _ii ≤ A _three ≤ ⋯ ≤ B ₃ ≤ B _ii ≤ B ₁ and |A_north − B_n | → 0 as northward → ∞. It follows that both sequences are Cauchy and have the same limit 50 , which must be the to the lowest degree upper jump for S .

Applications [edit]

The least-upper-spring holding of R can exist used to prove many of the main foundational theorems in real assay.

Intermediate value theorem [edit]

Permit f : [a, b] → R exist a continuous function, and suppose that f (a) < 0 and f (b) > 0. In this case, the intermediate value theorem states that f must have a root in the interval [a, b]. This theorem tin can exist proved by considering the set

S =  {south ∈ [a, b]  : f (x) < 0 for all x ≤ s} .

That is, Due south is the initial segment of [a, b] that takes negative values under f . Then b is an upper bound for South , and the to the lowest degree upper bound must be a root of f .

Bolzano–Weierstrass theorem [edit]

The Bolzano–Weierstrass theorem for R states that every sequence x_north of real numbers in a closed interval [a, b] must have a convergent subsequence. This theorem can be proved by considering the set

S =  {south ∈ [a, b]  : s ≤ x_n for infinitely many n}

Conspicuously, ${\displaystyle a\in South}$ , and South is not empty. In improver, b is an upper bound for South , and then Southward has a least upper leap c . And then c must be a limit point of the sequence 10_n , and it follows that ten_{due north} has a subsequence that converges to c .

Farthermost value theorem [edit]

Let f : [a, b] → R be a continuous function and allow Thousand = sup f ([a, b]), where M = ∞ if f ([a, b]) has no upper bound. The extreme value theorem states that M is finite and f (c) = M for some c ∈ [a, b]. This can be proved by because the set up

S =  {s ∈ [a, b]  :  sup f ([south, b]) = G} .

By definition of M , a ∈ Due south , and by its own definition, Due south is bounded by b . If c is the least upper bound of S , then information technology follows from continuity that f (c) = M .

Heine–Borel theorem [edit]

Let [a, b] be a closed interval in R , and let {U_α } exist a drove of open sets that covers [a, b]. And so the Heine–Borel theorem states that some finite subcollection of {U_α } covers [a, b] equally well. This argument can be proved by considering the set

Southward =  {due south ∈ [a, b]  :  [a, s] tin can exist covered past finitely many U_α } .

The set Southward obviously contains a , and is divisional past b past construction. By the least-upper-bound property, Due south has a to the lowest degree upper bound c ∈ [a, b]. Hence, c is itself an element of some open set up U_α , and it follows for c < b that [a, c + δ] can be covered past finitely many U_α for some sufficiently minor δ > 0. This proves that c + δ ∈ S and c is not an upper bound for S . Consequently, c = b .

History [edit]

The importance of the to the lowest degree-upper-bound property was first recognized by Bernard Bolzano in his 1817 paper Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, dice ein entgegengesetztes Resultat gewäahren, wenigstens eine reelle Wurzel der Gleichung liege.^[three]

See also [edit]

• List of real analysis topics

Notes [edit]

1. ^ Bartle and Sherbert (2011) define the "completeness belongings" and say that it is also called the "supremum belongings". (p. 39)
2. ^ Willard says that an ordered space "X is Dedekind complete if every subset of X having an upper bound has a to the lowest degree upper bound." (pp. 124-5, Problem 17E.)
3. ^ Raman-Sundström, Manya (August–September 2015). "A Pedagogical History of Firmness". American Mathematical Monthly. 122 (7): 619–635. arXiv:1006.4131. doi:10.4169/amer.math.monthly.122.vii.619. JSTOR x.4169/amer.math.monthly.122.vii.619. S2CID 119936587.

References [edit]

• Abbott, Stephen (2001). Agreement Assay. Undergraduate Texts in Mathematics. New York: Springer-Verlag. ISBN0-387-95060-5.
• Aliprantis, Charalambos D; Burkinshaw, Owen (1998). Principles of real analysis (Third ed.). Bookish. ISBN0-12-050257-vii.

• Bartle, Robert G.; Sherbert, Donald R. (2011). Introduction to Existent Analysis (iv ed.). New York: John Wiley and Sons. ISBN978-0-471-43331-6.
• Bressoud, David (2007). A Radical Arroyo to Real Assay. MAA. ISBN978-0-88385-747-two.
• Browder, Andrew (1996). Mathematical Analysis: An Introduction. Undergraduate Texts in Mathematics. New York: Springer-Verlag. ISBN0-387-94614-four.
• Dangello, Frank; Seyfried, Michael (1999). Introductory Existent Assay. Brooks Cole. ISBN978-0-395-95933-6.
• Rudin, Walter (1976). Principles of Mathematical Analysis . Walter Rudin Student Series in Advanced Mathematics (3 ed.). McGraw–Hill. ISBN978-0-07-054235-8.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN9780486434797.

Source: https://en.wikipedia.org/wiki/Least-upper-bound_property

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