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Completeness of the real numbers

The completeness of the real numbers is a fundamental property in mathematics, stating that every non-empty subset of the real numbers that is bounded above possesses a least upper bound (supremum) within the set of real numbers itself.^[1] This axiom ensures that the real line has no gaps, distinguishing the reals from incomplete ordered fields like the rational numbers, where subsets such as { r \in \mathbb{Q} \mid r^2 < 2 } are bounded above but lack a least upper bound in the rationals (namely \sqrt{2}).^[2] Formally, an ordered field is complete if, for any non-empty subset A bounded above, there exists an element x in the field such that x is an upper bound for A and no smaller element serves as an upper bound.^[3] This property is essential for real analysis, underpinning the existence of limits, the intermediate value theorem, and the fundamental theorem of calculus by guaranteeing convergence and continuity in the real numbers.^[4] Without completeness, many core results of calculus would fail, as seen in the rational numbers, where sequences like the decimal approximations to \sqrt{2} converge but their limit does not exist within the rationals.^[2] The completeness axiom implies other key features, such as the Archimedean property, which states that for any positive real a and any real b, there exists a natural number n such that na > b, ensuring the reals are not "too dense" in a pathological way.^[1] Equivalently, completeness can be expressed in terms of Dedekind cuts, where each real number corresponds to a partition of the rationals into two non-empty sets A and B such that all elements of A are less than all in B, and A has no greatest element; the reals are then constructed as the set of all such cuts to achieve completeness.^[3] Another formulation is Cauchy completeness: every Cauchy sequence of real numbers converges to a limit in the reals, which is crucial for metric spaces and topology.^[4] These axioms collectively define the real numbers as the unique (up to isomorphism) complete ordered field, providing the rigorous foundation for advanced mathematics.^[1]

Motivation and Background

Incompleteness of the rational numbers

In an ordered field, a gap arises in a nonempty subset that is bounded above but lacks a least upper bound within the field itself.^[5] The rational numbers \mathbb{Q}, despite forming an ordered field, contain such gaps, demonstrating their incompleteness as an ordered structure.^[5] A prominent example is the set S = \{ q \in \mathbb{Q} \mid q^2 < 2 \}. This set is nonempty, since $1 \in S, and bounded above, for example by $2. Yet, S possesses no least upper bound in \mathbb{Q}.^[5] If a least upper bound b \in \mathbb{Q} existed, then b^2 = 2 would hold, but \sqrt{2} is irrational, leading to a contradiction: assuming \sqrt{2} = p/q in lowest terms implies p^2 = 2q^2, forcing both p and q even, violating the lowest terms condition.^[5] Consequently, any purported rational upper bound exceeds \sqrt{2} or allows a smaller rational upper bound still greater than elements of S, preventing a least one.^[6] This phenomenon extends to other algebraic irrationals. For instance, the set \{ q \in \mathbb{Q} \mid q^3 < 2 \} is bounded above but lacks a least upper bound in \mathbb{Q}, as

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is irrational: assuming

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in lowest terms yields p^3 = 2q^3, implying p and q share a common factor of 2 by unique factorization, a contradiction.^[7] More generally, roots of irreducible polynomials with integer coefficients and no rational roots—identified via the rational root theorem, which limits possible rational roots to factors of the constant term over leading coefficient—produce similar gaps in \mathbb{Q}. Such gaps cause key analytic theorems to fail over \mathbb{Q}. The intermediate value theorem, which guarantees continuous functions attain all values between endpoints, breaks down for rational-valued functions; for example, define f: \mathbb{Q} \to \mathbb{Q} by f(x) = -1 if x^2 < 2 and f(x) = 1 if x^2 > 2, so f(1) = -1 and f(2) = 1, yet f never equals 0 on \mathbb{Q} due to the gap at \sqrt{2}.^[8] The real numbers remedy these issues by satisfying the least upper bound property.^[5]

Historical development

The concept of completeness in the real numbers traces its origins to ancient Greek mathematics, where early efforts to handle irrational quantities laid foundational groundwork. In the 4th century BCE, Eudoxus of Cnidus developed the method of exhaustion, a technique for approximating areas and volumes by inscribing and circumscribing polygons that converge to the true value, effectively addressing the existence of irrationals without explicitly constructing a complete number system.^[9]^[10] This approach, later formalized in Euclid's Elements, provided a rigorous way to deal with quantities like the square root of 2 by bounding them between rationals, foreshadowing the need for a continuum that fills all "gaps" in the rationals.^[11] The 19th century marked a pivotal shift toward rigorous foundations for analysis, driven by the recognition of gaps in rational numbers, such as the failure of certain sequences to converge within them. In 1817, Bernard Bolzano published his Rein analytischer Beweis, introducing a precise definition of continuity for functions and proving the intermediate value theorem using the concept of bounded sets with no maximum element, which implicitly relied on the density and completeness of the reals.^[12]^[13] Augustin-Louis Cauchy advanced this rigor in his 1821 Cours d'analyse, where he defined limits through inequalities involving arbitrary small quantities (precursors to epsilon-delta) and introduced Cauchy sequences as a tool for convergence, emphasizing the need for a number system where such sequences always converge.^[14]^[15] Richard Dedekind provided a constructive resolution in 1872 with Stetigkeit und irrationale Zahlen, defining real numbers as partitions (cuts) of the rationals and explicitly stating the least upper bound property as the defining completeness axiom for the reals, ensuring no gaps remain after incorporating irrationals.^[16]^[17] Karl Weierstrass complemented this by formalizing continuity and limits via the epsilon-delta definition in his Berlin lectures from the 1850s onward, which underscored sequential completeness as essential for uniform convergence in analysis.^[18] Meanwhile, Georg Cantor contributed through set theory, proving the uncountability of the reals in 1874 using the nested interval principle and developing nested interval theorems that highlight completeness as a property distinguishing the reals from countable dense sets like the rationals.^[19]^[20] These developments collectively established modern real analysis on a complete, uncountable continuum.^[10]

Axiomatic Definitions

Least upper bound property

The least upper bound property, also known as the supremum property or completeness axiom, states that every nonempty subset of the real numbers that is bounded above has a least upper bound in the real numbers. Formally, for any nonempty set S \subseteq \mathbb{R} that is bounded above, there exists s \in \mathbb{R} such that s is an upper bound for S (i.e., x \leq s for all x \in S) and s is the least such upper bound (i.e., for every \varepsilon > 0, there exists y \in S with s - \varepsilon < y \leq s).^[21] An upper bound of a set S \subseteq \mathbb{R} is any real number greater than or equal to every element in S, while the least upper bound, or supremum (denoted \sup S), is the smallest such upper bound. For example, consider the set S = \{ 1 - \frac{1}{n} \mid n \in \mathbb{N} \}. Here, 1 is an upper bound because $1 - \frac{1}{n} \leq 1 for all natural numbers n, and it is the least upper bound since for any \varepsilon > 0, choosing n > \frac{1}{\varepsilon} yields $1 - \frac{1}{n} > 1 - \varepsilon. This property ensures that the real numbers fill all "gaps" in the number line without leaving any bounded nonempty sets without a supremum.^[21] This axiom implies the Archimedean property of the real numbers, which states that for any positive real numbers x and y, there exists a natural number n such that n x > y. To sketch the proof, first note that the set of natural numbers \mathbb{N} (including 0) has no upper bound in \mathbb{R}; if it did, say with least upper bound \alpha, then \alpha - 1 would not be an upper bound, so some k \in \mathbb{N} satisfies \alpha - 1 < k \leq \alpha, implying k + 1 > \alpha, a contradiction. Thus, for any y/x > 0, there exists n \in \mathbb{N} with n > y/x, so n x > y.^[22] The least upper bound property distinguishes the real numbers from the rational numbers, as \mathbb{Q} lacks this property for certain sets. For instance, the set T = \{ q \in \mathbb{Q} \mid q^2 < 2 \} is nonempty and bounded above in \mathbb{Q} (e.g., by 2), but it has no least upper bound in \mathbb{Q}, since its supremum is \sqrt{2}, which is irrational. However, any ordered field satisfying the least upper bound property is isomorphic to \mathbb{R}, meaning that assuming this axiom in an ordered field constructs the real numbers up to isomorphism.^[21]^[23]

Dedekind completeness

A Dedekind cut of the rational numbers \mathbb{Q} is a partition of \mathbb{Q} into two nonempty subsets A and B such that A \cup B = \mathbb{Q}, A \cap B = \emptyset, every element of A is less than every element of B, and A has no greatest element.^[24]^[25] The set of real numbers \mathbb{R} is constructed as the collection of all Dedekind cuts, where each cut represents a real number. The order on \mathbb{R} is defined by set inclusion: for two cuts (A, B) and (C, D), (A, B) < (C, D) if and only if A \subsetneq C. Arithmetic operations are defined componentwise on the lower sets: the sum of cuts (A, B) and (C, D) has lower set \{a + c \mid a \in A, c \in C\}, the product for positive cuts has lower set \{q \in \mathbb{Q} \mid q \leq r \cdot s for some r \in A, s \in C with r, s > 0\}, and additive inverses swap the lower and upper sets after negating rationals. This structure embeds \mathbb{Q} into \mathbb{R} via cuts like the one for q \in \mathbb{Q} with lower set \{r \in \mathbb{Q} \mid r < q\}.^[24]^[25] This construction yields a complete ordered field because every Dedekind cut corresponds uniquely to a real number, ensuring that the reals fill all gaps in the rationals; for any nonempty subset S \subseteq \mathbb{R} bounded above, the union of the lower sets of cuts in S forms the lower set of a new cut that serves as the least upper bound of S.^[24]^[25] Dedekind completeness—the property that every Dedekind cut defines a unique real number—is equivalent to the least upper bound property. To see this, assume the least upper bound property holds; for any partition of \mathbb{R} into lower set H and upper set K with all elements of H less than those of K, the least upper bound w of H must be the first element of K (or last of H, but the no-maximum condition prevents this), completing the cut. Conversely, assuming Dedekind completeness, for any nonempty S \subseteq \mathbb{R} bounded above, the set of non-upper bounds forms a lower set whose cut yields the least upper bound as the minimal upper bound.^[26] For example, the irrational number \sqrt{2} is represented by the Dedekind cut with lower set A = \{q \in \mathbb{Q} \mid q < 0 \lor q^2 < 2\} and upper set B = \mathbb{Q} \setminus A; this cut has no rational maximum in A and fills the gap where no rational squares to 2, demonstrating how the construction completes \mathbb{Q}.^[24]^[25]

Metric and Sequential Perspectives

Cauchy sequences and their convergence

In a metric space (X, d), a sequence (x_n)_{n=1}^\infty in X is a Cauchy sequence if, for every \epsilon > 0, there exists a positive integer N such that d(x_m, x_n) < \epsilon whenever m, n > N.^[27] This condition ensures that the terms of the sequence eventually become arbitrarily close to one another, regardless of the existence of a limit point in X. The concept originates from Augustin-Louis Cauchy's foundational work on limits, where he emphasized successive values approaching each other indefinitely without presupposing a fixed limit.^[27] In the real numbers \mathbb{R} with the standard metric d(x, y) = |x - y|, the definition simplifies to: for every \epsilon > 0, there exists N \in \mathbb{N} such that |x_m - x_n| < \epsilon for all m, n > N./03:_Vector_Spaces_and_Metric_Spaces/3.13:_Cauchy_Sequences._Completeness) This metric perspective allows Cauchy sequences to characterize convergence independently of the ambient space's structure, making them a powerful tool for studying completeness. The Cauchy criterion is equivalently stated as

\lim_{m,n \to \infty} |x_m - x_n| = 0,

capturing the uniform "clustering" of tail terms./03:_Vector_Spaces_and_Metric_Spaces/3.13:_Cauchy_Sequences._Completeness) A fundamental property is that every convergent sequence in any metric space is Cauchy. To see this, suppose (x_n) converges to some L \in X. Given \epsilon > 0, there exists N such that d(x_k, L) < \epsilon/2 for all k > N. Then, for m, n > N,

d(x_m, x_n) \leq d(x_m, L) + d(L, x_n) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.

This holds in \mathbb{R} with the absolute value metric./03:_Vector_Spaces_and_Metric_Spaces/3.13:_Cauchy_Sequences._Completeness) The converse—that every Cauchy sequence converges—requires the space to be complete, as in \mathbb{R}, but fails in incomplete spaces like the rationals \mathbb{Q}. For instance, the sequence of decimal approximations to \sqrt{2} (e.g., 1, 1.4, 1.41, 1.414, ...) is Cauchy in \mathbb{Q} under the standard metric but converges only to the irrational \sqrt{2} \notin \mathbb{Q}.^[28] Examples illustrate these properties clearly. The partial sums s_n = \sum_{k=0}^n r^k of a geometric series with |r| < 1 form a Cauchy sequence in \mathbb{R}, as the terms approach the limit $1/(1-r) and thus satisfy the criterion by the forward implication./09:_Sequences_and_Series/9.02:_Infinite_Series) In contrast, the partial sums s_n = \sum_{k=1}^n 1/k of the harmonic series are not Cauchy, since the sequence diverges to \infty; specifically, s_{2n} - s_n = \sum_{k=1}^n 1/(n+k) > n \cdot (1/(2n)) = 1/2, so the differences remain bounded away from zero for suitable m, n./09:_Sequences_and_Series/9.03:_The_Divergence_and_Integral_Tests) These cases highlight how Cauchy sequences detect convergence in complete spaces like \mathbb{R} without relying on explicit limits.

Cauchy completeness

The Cauchy completeness of the real numbers asserts that every Cauchy sequence in \mathbb{R} converges to a limit in \mathbb{R}.^[29] This property distinguishes \mathbb{R} from incomplete metric spaces and forms a cornerstone of real analysis. A sequence \{x_n\} in \mathbb{R} is Cauchy if for every \epsilon > 0, there exists N \in \mathbb{N} such that |x_m - x_n| < \epsilon for all m, n > N. The completeness theorem guarantees that such sequences always possess a limit within \mathbb{R}, ensuring that \mathbb{R} has no "holes" with respect to this convergence criterion.^[30] To outline the proof, first note that any Cauchy sequence in \mathbb{R} is bounded, as the terms eventually lie within an interval of length $2\epsilon for sufficiently small \epsilon > 0. By the Bolzano-Weierstrass theorem, this bounded sequence admits a convergent subsequence \{x_{n_k}\} converging to some L \in \mathbb{R}. To show the entire sequence converges to L, fix \epsilon > 0 and choose N_1 such that |x_m - x_n| < \epsilon/2 for m, n > N_1, and K such that |x_{n_k} - L| < \epsilon/2 for k > K. Select n > \max(N_1, n_K); then |x_n - L| \leq |x_n - x_{n_K}| + |x_{n_K} - L| < \epsilon. Thus, \{x_n\} \to L. (Details of the Bolzano-Weierstrass theorem are deferred to subsequent sections.)^[29] In the broader context of ordered fields, Cauchy completeness is equivalent to the least upper bound property, provided the field is Archimedean (meaning that for any x > 0 and y \in \mathbb{R}, there exists n \in \mathbb{N} with nx > y). Specifically, an Archimedean ordered field is order complete if and only if it is Cauchy complete.^[31] This equivalence underscores why \mathbb{R}, as the unique complete ordered field up to isomorphism, satisfies both formulations of completeness. In contrast, the rational numbers \mathbb{[Q](/page/Q)} lack Cauchy completeness. Consider the sequence of decimal approximations to \sqrt{2}, such as x_1 = 1, x_2 = 1.4, x_3 = 1.41, x_4 = 1.414, and so on, where each x_n is the truncation of \sqrt{2} to n decimal places. This sequence is Cauchy in \mathbb{[Q](/page/Q)} because the terms get arbitrarily close as n increases, yet it converges to \sqrt{2} \notin \mathbb{[Q](/page/Q)}, so no limit exists in \mathbb{[Q](/page/Q)}.^[32] Cauchy completeness provides the foundational basis for defining continuity, uniform convergence, and limits in real analysis using the metric structure of \mathbb{R}, independent of coordinate representations. It enables the development of key analytical tools, such as the extension of continuous functions to compact sets and the rigorous treatment of derivatives and integrals, without reliance on incomplete approximations.^[33]

Key Consequences and Theorems

Nested intervals theorem

The nested intervals theorem is a fundamental consequence of the completeness of the real numbers, providing a criterion for the existence of a unique point in the intersection of a decreasing sequence of closed intervals whose lengths contract to zero. Specifically, consider a sequence of closed intervals \{I_n = [a_n, b_n]\}_{n=1}^\infty in \mathbb{R} such that I_{n+1} \subseteq I_n for all n \in \mathbb{N} (i.e., a_n \leq a_{n+1} \leq b_{n+1} \leq b_n) and the length b_n - a_n \to 0 as n \to \infty. Then, the intersection \bigcap_{n=1}^\infty I_n consists of exactly one point.^[34]^[35] To prove this using the Cauchy completeness of \mathbb{R}, first observe that the sequence (a_n) is non-decreasing and bounded above (e.g., by b_1), so it is bounded. Moreover, for any \epsilon > 0, there exists N such that b_n - a_n < \epsilon for all n \geq N; thus, for m, n \geq N with m \geq n, |a_m - a_n| \leq b_n - a_n < \epsilon, showing that (a_n) is a Cauchy sequence. By the completeness of \mathbb{R}, a_n \to x for some x \in \mathbb{R}. Similarly, (b_n) is non-increasing and bounded below, hence also Cauchy and converges to the same limit x. Since a_n \leq x \leq b_n for all n (as x is the limit of both endpoints), it follows that x \in I_n for every n, so x \in \bigcap_{n=1}^\infty I_n. For uniqueness, suppose y \in \bigcap_{n=1}^\infty I_n with y \neq x; then |x - y| > 0, but for sufficiently large n, b_n - a_n < |x - y|, contradicting y \in [a_n, b_n] and x \in [a_n, b_n].^[35]^[36] A classic example illustrates the theorem through the bisection method for constructing \sqrt{2}. Begin with the initial interval I_1 = [0, 2], noting that $0^2 = 0 < 2 and $2^2 = 4 > 2. The midpoint is 1, and since $1^2 = 1 < 2, set I_2 = [1, 2]. Repeat: the midpoint of I_2 is 1.5, and $1.5^2 = 2.25 > 2, so I_3 = [1, 1.5]. Continuing this process, each I_n is closed, nested, and has length $2^{-n+1} \to 0, so \bigcap_{n=1}^\infty I_n = \{\sqrt{2}\}, confirming the existence and uniqueness of the positive square root of 2 in [0, 2].^[35]^[37] The theorem generalizes beyond intervals to nested sequences of nonempty compact subsets of \mathbb{R} whose diameters tend to zero: their intersection contains exactly one point. In \mathbb{R}, compact sets are precisely the closed and bounded ones by the Heine-Borel theorem, allowing the nested intervals argument to extend via finite subcovers and diameter control.^[38]^[36]

Monotone convergence theorem

The monotone convergence theorem states that every monotone and bounded sequence of real numbers converges to a limit in the real numbers. Specifically, if a sequence \{x_n\} in \mathbb{R} is increasing (i.e., x_{n+1} \geq x_n for all n) and bounded above, then it converges to its least upper bound \sup\{x_n : n \in \mathbb{N}\}; similarly, if it is decreasing and bounded below, it converges to its greatest lower bound \inf\{x_n : n \in \mathbb{N}\}.^[39] To prove the theorem for an increasing sequence bounded above by some M \in \mathbb{R}, let L = \sup\{x_n : n \in \mathbb{N}\}, which exists by the least upper bound property of the reals. For any \epsilon > 0, L - \epsilon is not an upper bound of the set \{x_n\}, so there exists N \in \mathbb{N} such that x_N > L - \epsilon. Since the sequence is increasing, x_n \geq x_N > L - \epsilon for all n \geq N. Also, x_n \leq L for all n, so |x_n - L| < \epsilon for all n \geq N, proving \lim_{n \to \infty} x_n = L. The proof for decreasing sequences is analogous.^[39] In equation form, for an increasing bounded sequence \{x_n\},

\lim_{n \to \infty} x_n = \sup\{x_n : n \in \mathbb{N}\}.

^[39] A classic example is the sequence generated by Newton's method for approximating \sqrt{a} where a > 0. Starting with x_1 > 0, define x_{n+1} = \frac{1}{2} \left( x_n + \frac{a}{x_n} \right). If $0 < x_1 \leq \sqrt{a}, the sequence is increasing and bounded above by \sqrt{a}, hence converges to \sqrt{a} by the monotone convergence theorem. For instance, with a = 2 and x_1 = 1, the terms increase toward \sqrt{2} \approx 1.414213562.^[40]

Bolzano–Weierstrass theorem

The Bolzano–Weierstrass theorem asserts that every bounded sequence in \mathbb{R} has a convergent subsequence.^[41] This result highlights a key aspect of the completeness of the real numbers, ensuring that boundedness alone guarantees the existence of a limit point for the sequence, even if the full sequence does not converge.^[42] To outline the proof, consider a bounded sequence \{a_n\} with |a_n| \leq M for all n. Divide the interval [-M, M] into two subintervals of equal length and select one that contains infinitely many terms of the sequence. Repeat this bisection process indefinitely, yielding a nested sequence of closed intervals I_k = [c_k, d_k] where each I_k contains infinitely many terms, and the length d_k - c_k = 2M / 2^k \to 0 as k \to \infty. By the nested intervals theorem, the intervals intersect at a unique point L \in \mathbb{R}. Selecting one term a_{n_k} from each I_k with n_1 < n_2 < \cdots produces a subsequence \{a_{n_k}\} that converges to L, since for any \varepsilon > 0, eventually I_k \subset (L - \varepsilon, L + \varepsilon).^[43] For illustration, the sequence a_n = (-1)^n n is unbounded, as |a_n| = n \to \infty, and thus lacks any convergent subsequence.^[41] Conversely, the bounded sequence a_n = \sin n (with n in radians) admits convergent subsequences; for instance, certain subsequences approach 1 or -1, reflecting the dense distribution of \{n \mod 2\pi\} in [0, 2\pi).^[42] This theorem is equivalent to the Heine–Borel characterization of compactness in \mathbb{R}, which states that a subset is compact if and only if it is closed and bounded; the Bolzano–Weierstrass property ensures sequential compactness for such sets.^[44] Historically, the result traces to Bernhard Bolzano's 1817 work on the intermediate value theorem, where he invoked a precursor lemma on bounded sequences having convergent subsequences, though published obscurely; Karl Weierstrass later provided a rigorous proof in 1854 while establishing foundations of analysis.^[45]

Intermediate value theorem

The intermediate value theorem states that if f: [a, b] \to \mathbb{R} is a continuous function with f(a) < c < f(b), then there exists some x \in (a, b) such that f(x) = c.^[46] To prove this using the least upper bound property, define the set S = \{ x \in [a, b] \mid f(x) \leq c \}. This set is nonempty since it contains a, and it is bounded above by b, so by the least upper bound property of the real numbers, S has a supremum x_0 = \sup S \in [a, b].^[46] Suppose f(x_0) > c; then by continuity of f at x_0, there exists \delta > 0 such that f(x) > c for all x \in (x_0 - \delta, x_0 + \delta) \cap [a, b]. In particular, for points x < x_0 sufficiently close to x_0, f(x) > c, so these points are not in S. However, since x_0 = \sup S, there must be elements of S arbitrarily close to x_0 from below, leading to a contradiction. Similarly, supposing f(x_0) < c implies, by continuity, a neighborhood around x_0 where f(x) < c, so x_0 \in S and points x > x_0 nearby are also in S, contradicting that x_0 is the supremum of S. Thus, f(x_0) = c, and since f(a) < c < f(b), it follows that x_0 \in (a, b).^[46]^[47] A classic example is the function f(x) = x^2 - 2 on the interval [1, 2], where f(1) = -1 < 0 < 2 = f(2), so by the intermediate value theorem, there exists x \in (1, 2) such that f(x) = 0, namely x = \sqrt{2}.^[46] This theorem extends the completeness of the real numbers to the connectedness of \mathbb{R}, as the least upper bound property ensures that \mathbb{R} cannot be partitioned into two nonempty disjoint open sets, making continuous images of intervals into \mathbb{R} fill all intermediate values.^[47]

Open induction principle

The open induction principle serves as a completeness-based analog to mathematical induction for properties defined on the positive real numbers, enabling proofs where standard induction on natural numbers falls short due to the uncountable nature of the reals. It asserts that if a property P holds for all x > 0 except possibly on a gapless initial segment—meaning the exceptional set lacks gaps and is connected from the origin—then P holds for all x > 0. Formally, if the set E = \{ x > 0 \mid \neg P(x) \} is bounded above or empty, and the set where P holds is open and inductive (i.e., if P(y) for all y < x, then P(x)), then P(x) holds for all x > 0.^[48] The proof proceeds via the least upper bound property of the reals. Suppose E is nonempty and bounded above; let s = \sup E. If s is finite, then since the set where P holds is open, s \notin E, because if s \in E, there would be points greater than s in E by the inductive property, contradicting the supremum. All y < s satisfy P(y) (since any counterexample would lower the supremum), so the inductive property implies P(s). This leads to the conclusion that E must be empty. If E is unbounded above, it contradicts the boundedness assumption.^[48] A representative example is the proof that every positive real number has a positive integer part, or floor function. Consider the property P(x): there exists a positive integer n such that n \leq x < n+1. The set where \neg P(x) is empty. To apply the principle, note that for x > 0, the set S = \{ n \in \mathbb{N} \mid n \leq x \} is nonempty (by the Archimedean property, derived from completeness) and bounded above by x, so s = \sup S exists. Let n be the greatest integer less than or equal to s. Then n \in S (since s is the sup of integers), and n+1 > s, so n \leq x < n+1. This demonstrates P(x) for all x > 0, filling the gap left by the lack of well-ordering on the reals beyond the naturals.^[49] This principle relates to the well-ordering of the natural numbers by extending inductive reasoning to the dense reals, where direct well-ordering fails, using completeness to "close gaps" in initial segments without requiring discrete successors.^[48]

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[PDF] Who Gave You the Epsilon? Cauchy and the Origins of Rigorous ...
Cauchy systematically translated this refined limit-concept into the algebra of inequalities, and used it in proofs once it had been so translated; thus he gave ...
[15]
Cauchy's Calculus - MacTutor History of Mathematics
Cauchy wrote Cours d'Analyse (1821) based on his lecture course at the École Polytechnique. In it he attempted to make calculus rigorous.Missing: introduction sequences
[16]
Dedekind's Contributions to the Foundations of Mathematics
Apr 22, 2008 · ... Dedekind-cuts themselves as “the real numbers”. While aware of this option (Dedekind 1876b, 1888b), Dedekind tells us to apply “abstraction ...
[17]
10.2: Building the Real Numbers - Mathematics LibreTexts
May 28, 2023 · The method of “Dedekind cuts” first developed by Richard Dedekind (though he just called them “cuts”) in his 1872 book, Continuity and the ...
[18]
[PDF] On the history of epsilontics - arXiv
It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass' definition of a limit. The article gives various ...Missing: completeness | Show results with:completeness
[19]
The Early Development of Set Theory
Apr 10, 2007 · The Cantor and Dedekind definitions of the real numbers relied implicitly on set theory, and can be seen in retrospect to involve the ...
[20]
[PDF] On Cantor's uncountability proofs - arXiv
of all real numbers is uncountable, i.e. that any bijection Ù ¨ — is impossible [1, 2]. He ...
[21]
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/01%3A_Tools_for_Analysis/1.05%3A_The_Completeness_Axiom_for_the_Real_Numbers
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[PDF] The Archimedean Property - Penn Math
Sep 3, 2014 · Since N ⊂ R and R has the least upper bound property, then N has a least upper bound α ∈ R. Thus n ≤ α for all n ∈ N and is the smallest such ...
[23]
[PDF] Uniqueness of real numbers - Williams College
ABSTRACT. We prove that any two complete ordered fields are isomorphic to one another. Put differently: R is the only complete ordered field (up to isomorphism) ...
[24]
[PDF] Dedekind Cuts - Brown Math
Sep 17, 2014 · So, if you define a real number to be a Dedekind cut then the set R of real numbers is a complete ordered field with the least upper bound ...
[25]
[PDF] CONSTRUCTION OF THE REAL NUMBERS
We present a brief sketch of the construction of R from Q using Dedekind cuts. This is the same approach used in Rudin's book Principles of Mathematical ...
[26]
[PDF] The least upper bound principle is equivalent to the Dedekind cut ...
By the Dedekind property, either H has a last element or K has a first element. If H had a last element, say w, then w would also be an upper bound of H, a.Missing: completeness | Show results with:completeness
[27]
[PDF] Cauchy's Cours d'analyse
... 1821, Cauchy had published 28 memoirs, but the Cours d'analyse was his first full-length book. Cauchy's first original mathematics concerned the geometry of ...
[28]
[PDF] Cauchy's Construction of R - UCSD Math
If (an) is a convergent rational sequence (that is, an → q for some rational number q), then (an) is a Cauchy sequence. Proof. We know that an → q. Here ...
[29]
[PDF] 18.100A Fall 2020 Lecture 10: The Completeness of the Real ...
Theorem 7. A sequence of real numbers {xn} is Cauchy if and only if {xn} is convergent. Proof: ( =⇒ ) If {xn} is Cauchy, then {xn} is bounded. Therefore ...
[30]
2.4 Cauchy sequences - Penn Math
Theorem 2.4.8. Every Cauchy sequence of real numbers converges. 🔗. 🔗. 🔗. Subsection 2.4.2 Decimal representations. In the "real world" we normally interact ...<|control11|><|separator|>
[31]
[PDF] Thm 1.4.3. Order Complete → Cauchy Complete & Archimedean
Feb 7, 2024 · Theorem 1.4. 3. An ordered field is order complete if and only if it is Cauchy complete and Archimedean. natural number N such that for all n > ...
[32]
[PDF] Math 320-2: Real Analysis - Northwestern University, Lecture Notes
More generally, any sequence of rationals which converges to an irrational will be an example of Cauchy sequence in Q which does not converge in Q. This ...
[33]
[PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
A sequence of real numbers converges if and only if it is a Cauchy sequence. The fact that real Cauchy sequences have a limit is an equivalent way to formu-.
[34]
[PDF] 1.6 The Nested Intervals Theorem
The Nested Intervals Theorem states that if closed intervals are nested and their lengths converge to 0, then the intersection of these intervals contains ...
[35]
[PDF] Real Analysis
Sep 26, 2015 · Proof of The Nested Interval Theorem 1.49. Let A = {an : n > 0} and let B = {bn : n > 0}. Since the intervals are non-empty and nested, an ...
[36]
[PDF] Topology of the Real Numbers - UC Davis Math
In the case when each Kn = [an,bn] is a compact interval, the preceding result is called the nested interval theorem. Example 5.44. The nested compact intervals ...
[37]
[PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeare's ...
[38]
[PDF] Chapter 5 Compactness
COMPACT SPACES AND SUBSPACES. 45 so the diameters of these intervals goes to zero. By the Cantor Nested Intervals Theorem, we know that there is precisely one.
[39]
[PDF] The Monotone Convergence Theorem - UMD MATH
The Monotone Convergence Theorem states that if a sequence is monotone and bounded, it converges. If increasing, it converges to sup{an}; if decreasing, to inf ...
[40]
[PDF] Square Roots via Newton's Method - MIT Mathematics
Feb 4, 2015 · Recall that Newton's method finds an approximate root of f(x)=0 from a guess xn by approximating f(x) as its tangent line f(xn) + f0(xn)(x − xn ...
[41]
[PDF] introduction to the convergence of sequences - UChicago Math
Jul 12, 2015 · Bolzano-Weierstrass Theorem: Every bounded sequence contains a convergent subsequence. Example 5.6. Given a sequence xn = (1,2,3,4,1,2,3,4...) ...
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[PDF] 2.3. Bolzano-Weierstrass Theorem - East Tennessee State University
Feb 5, 2024 · In this section we show that every bounded set of real numbers has a “limit point” in the Bolzano-Weierstrass Theorem (Theorem 2-12). We define ...
[43]
[PDF] Section 2.3: Bolzano-Weierstrass Theorem
Oct 19, 2011 · Theorem 2-12 (Bolzano-Weierstrass): Every bounded infinite set of real numbers has at least one limit point. Note: Clearly some bounded infinite ...Missing: historical | Show results with:historical
[44]
On the equivalence of the Heine-Borel and the Bolzano-Weierstrass ...
Specifically, the Heine-Borel theorem asserts that every bounded closed subset of R is compact, while Bolzano-Weierstrass states every bounded infinite subset ...
[45]
Bolzano's Intermediate Value Theorem
Weierstrass' proof in 1854 is the best known. Bolzano needed a lemma, that every bounded, infinite sequence of reals has a convergent subsequence. This, today, ...Missing: statement | Show results with:statement
[46]
Intermediate Value Theorem
Hence by the "completeness property" of the reals, the least upper bound, or supremum c = sup S exists. ... The next proof in this series is the Bounded Value ...
[47]
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Below is a merged response summarizing the connectedness of the real line, its relation to completeness, and the Intermediate Value Theorem based on the provided segments from Rudin's "Principles of Mathematical Analysis." To retain all information in a dense and organized manner, I will use a table in CSV format for key details, followed by a narrative summary that integrates the content. Since the system has a "no thinking token" limit, I’ll focus on directly compiling and structuring the provided data without additional inference or elaboration beyond what’s given.
[48]
[PDF] An Induction Principle over Real Numbers - Hal-Inria
Oct 4, 2016 · Proof of the open induction principle. In this section we give the complete proof of the open induction principle for enumerative open sets.
[49]
[PDF] i. introduction. elements of mathematical logic and set theory. real ...
Analogously, for a nonempty set N bounded from above, supremum of N,. supN defines as the least upper bound of N. ... Theorem 1.5 (Existence of integer part). (∀x ...<|control11|><|separator|>