Real Analysis Primer

The section begins by adopting a naive approach to set theory, stating that a set can be defined by collecting objects that share a common property. This property can be specified either through explicit enumeration (e.g., the set {a, b, c}) or by describing the characteristic property (e.g., the set of positive integers). This foundational explanation of sets lays the basis for later mathematical constructions. The text emphasizes that this is a non-axiomatic approach, focusing on intuitive understanding rather than rigorous axiomatic development. This contrasts with more advanced treatments of set theory but suffices for the introductory nature of real analysis within this context. The simplicity of this approach allows the reader to quickly grasp the fundamental concepts required to continue with the text. The implicit assumption is that the reader is familiar with basic set operations such as union, intersection, and subset relationships. Although these aren't explicitly detailed, they are presumed as prerequisites for understanding the material that follows. The overall objective is to provide a sufficiently clear and readily accessible understanding of sets to support the core concepts of real analysis.

The concept of equivalence relations is introduced, although the details are not extensively elaborated upon. The text assumes a prior familiarity with the definition and basic properties of equivalence relations. The reader is expected to understand that an equivalence relation partitions a set into equivalence classes. This is crucial for subsequent sections where equivalence relations might be implicitly used. The document does not delve into the formal properties of reflexivity, symmetry, and transitivity, but presupposes this knowledge as a prerequisite for following the later arguments. An exercise is presented ('Find the equivalence classes for the equivalence relation in Exercise 1.1.2'), indicating that equivalence relations are not simply introduced but are meant to be actively engaged with by the reader. The implication is that understanding equivalence classes is vital for grasping the concepts developed further in the document. The text's focus is on real analysis, not a comprehensive treatment of set theory, hence the brief introduction of this fundamental concept. The exercise highlights the importance of applying this theoretical concept for a better understanding of the overall text.

This subsection establishes the order and metric properties of rational numbers. A rational number 'a' is defined as positive if it can be expressed as p/q, where p and q are positive integers. The ordering of rational numbers is defined: a < b (or b > a) if b - a is positive. This naturally leads to the definitions of negative numbers (a < 0) and the less than or equal to (≤) and greater than or equal to (≥) relations. The section emphasizes that these definitions are fundamental to the structure of rational numbers and their subsequent use in constructing real numbers. The importance of the ordering is highlighted through the introduction of the concept of upper and lower bounds for sets of rational numbers. An exercise focusing on the uniqueness of the supremum (least upper bound) if it exists, further solidifies the understanding of these ordering concepts. This section establishes the basic arithmetic and order relationships for rational numbers, laying the foundation for more advanced concepts like supremum and infimum that are essential for understanding the limitations of the rational number system.

The concepts of upper and lower bounds are formally introduced for subsets of rational numbers. An upper bound 's' for a set A is a rational number such that s ≥ a for every a in A. The supremum (or least upper bound, sup A) is defined as an upper bound that is less than or equal to every other upper bound. The infimum (or greatest lower bound, inf A) is defined analogously for lower bounds. This formalization is crucial for understanding the completeness property of real numbers and how it differs from the rational numbers. The section emphasizes that not all subsets of rational numbers possess a supremum or infimum. A key proposition, 'There does not exist a rational number s with the property that s² = 2,' demonstrates a fundamental limitation of rational numbers and strongly motivates the need for constructing a more complete number system—the real numbers. This proposition is a cornerstone in the development of the concept of real numbers in the subsequent sections. The clear definition of supremum and infimum sets the stage for examining their existence within different number systems. This difference is central to understanding the need to extend the rational numbers to the real numbers.

This subsection introduces sequences of rational numbers. A sequence is defined as a function from a subset of integers (typically {n, n+1, n+2,...}) to a set A. The notation for sequences is established, using both functional notation (ϕ(i)) and subscript notation (aᵢ). The section provides examples to illustrate these notations clearly, making it easy for the reader to grasp the concepts. The focus here is to provide a clear and concise definition of sequences and establish the notation used throughout the rest of the document. While seemingly simple, this foundational understanding is critical for the later sections, particularly the construction of real numbers using Cauchy sequences. The section provides a very straightforward introduction to the topic, setting the groundwork for more complex concepts that will be introduced further in the document. This foundational understanding is necessary before transitioning to more complex definitions and concepts that are integral to the construction of real numbers.

This section presents a construction of the real numbers using Cauchy sequences of rational numbers. It begins by defining the set C as the set of all Cauchy sequences of rational numbers. A crucial step is defining an equivalence relation ∼ on C. Two Cauchy sequences {aᵢ} and {bⱼ} are equivalent ({aᵢ} ∼ {bⱼ}) if, for every rational number ε > 0, there exists an integer N such that |aᵢ - bᵢ| < ε for all i > N. This equivalence relation is fundamental because it groups Cauchy sequences that represent the same real number. The equivalence classes formed by this relation then define the real numbers. This construction directly addresses the incompleteness of the rational numbers. The rational numbers lack elements that are intuitively expected to exist, such as the square root of 2. This method ensures that all Cauchy sequences converge to a limit within the real numbers, thereby eliminating the gaps present in the rationals. This equivalence relation is carefully constructed to ensure that the resulting set of real numbers is complete. The detailed definition of equivalence and the careful explanation of the rationale behind this construction are essential for building a thorough understanding of the structure of real numbers.

Having defined real numbers as equivalence classes of Cauchy sequences, the section proceeds to define arithmetic operations (addition and multiplication) on these real numbers. Addition and multiplication of real numbers are defined in terms of the corresponding operations on the Cauchy sequences representing them. It's shown that these operations are well-defined, meaning the result doesn't depend on the choice of representative from the equivalence class. Similarly, an order relation is defined on the real numbers, declaring a real number u to be positive (u > 0) if it's represented by a Cauchy sequence {aᵢ} such that there's a rational ε > 0 and an integer N with aᵢ > ε for all i > N. This naturally leads to the definition of negative real numbers and the usual ordering relations (<, >, ≤, ≥). These definitions are key to establishing the algebraic and order properties of the real number system. This careful development ensures that the real numbers inherit the expected properties from the rational numbers while simultaneously addressing their limitations. The fact that these properties are explicitly defined and demonstrated to be well-defined is a critical aspect of the construction. The rigorous treatment of order provides the foundation for later discussions involving concepts like supremum and infimum.

This subsection demonstrates the completeness property of the real numbers, which is a crucial distinction from the rational numbers. The concepts of upper and lower bounds for sets of real numbers are defined analogously to their counterparts for rational numbers. The critical result is that every nonempty subset of real numbers that is bounded above has a supremum (least upper bound), and every nonempty subset bounded below has an infimum (greatest lower bound). This is the completeness property, directly addressing the deficiencies of the rational numbers. This property ensures that there are no 'gaps' in the real number line; every bounded set has a least upper and greatest lower bound within the real numbers. This section also highlights the uniqueness of the supremum and infimum when they exist. This fundamental difference between real and rational numbers provides justification for the construction presented in the preceding subsections and lays a solid foundation for many essential results in real analysis. The completeness property is central to many important theorems and concepts used in advanced mathematical studies. The careful demonstration of this property within the framework of Cauchy sequences is a significant contribution to understanding the nature of real numbers.

This section focuses on the convergence of sequences, particularly those that are monotonic and bounded. A sequence {aᵢ} is defined as nondecreasing if aᵢ ≤ aᵢ₊₁ for all i, and nonincreasing if aᵢ ≥ aᵢ₊₁ for all i. A sequence is bounded if there exists a real number M such that |aᵢ| ≤ M for all i. A key theorem states that if {aᵢ} is a nondecreasing and bounded sequence of real numbers, then {aᵢ} converges to its supremum (least upper bound). This theorem elegantly connects the concept of convergence with the completeness property of real numbers, highlighting the importance of the supremum and infimum concepts introduced earlier. The proof relies on the completeness property of the real numbers, demonstrating a direct connection between the structure of real numbers and the behavior of sequences. The theorem establishes a fundamental result about the convergence of monotonic bounded sequences, which is a crucial tool for proving many other results in real analysis. Exercises are included to extend this theorem to nonincreasing sequences, thereby establishing a more comprehensive understanding of the behavior of bounded monotonic sequences.

The section expands the discussion of sequences to include those that diverge to positive or negative infinity. A sequence {aᵢ} is said to diverge to positive infinity (limᵢ→∞ aᵢ = +∞) if for every real number M, there exists an integer N such that aᵢ > M whenever i > N. Divergence to negative infinity is defined similarly. This extends the concept of convergence to include unbounded sequences, providing a more complete framework for analyzing sequence behavior. The section shows that any nondecreasing sequence of real numbers either converges to a finite limit or diverges to positive infinity. This result provides a characterization of the possible limiting behavior of nondecreasing sequences, clarifying the relationship between boundedness and convergence. This subsection emphasizes that divergence to infinity is a distinct type of limiting behavior, and this expanded definition complements the earlier discussions on convergence to finite limits. The concepts of sequences diverging to infinity provide a more comprehensive understanding of the behavior of sequences, enhancing the analytical tools available in real analysis.

This section introduces the concept of cardinality, a way to compare the sizes of sets. A set A is defined as finite if it has the cardinality of the set {1, 2, 3, ..., n} for some positive integer n, denoted |A| = n. A set is defined as countable if it has the cardinality of the set of positive integers Z⁺, denoted |A| = ℵ₀. This establishes a fundamental distinction between finite and infinite sets, laying the groundwork for comparing the 'sizes' of infinite sets. The distinction between finite and countable sets is crucial, and this section clearly defines these concepts using standard set theory terminology. The use of the symbol ℵ₀ for the cardinality of countable sets is standard mathematical notation, which is introduced here to maintain consistency with established mathematical conventions. The definitions and notations established are critical for further discussions on cardinality in the following parts of the section, allowing for the comparison of different types of infinite sets.

The concept of a power set P(A) (the set of all subsets of A) is introduced. For finite sets, the cardinality of the power set is shown to be 2ⁿ, where n is the cardinality of the original set. This relationship between the cardinality of a set and its power set is extended to infinite sets, defining |P(A)| = 2^|A| for any nonempty set A. The main result of this section is Cantor's Theorem, which states that for any nonempty set A, |A| < |P(A)|. This theorem demonstrates that the cardinality of the power set of A is strictly greater than the cardinality of A, regardless of whether A is finite or infinite. The proof of Cantor's theorem utilizes a diagonalization argument, which is a standard proof technique in set theory. The theorem is presented concisely, focusing on the key idea that the power set of any set is always 'larger' than the original set, leading to the conclusion that there are different 'sizes' of infinity. This result has profound implications in mathematics, particularly in set theory and analysis. The clear presentation of the proof and the explanation of its significance are essential components of this section.

This section begins by defining the limit of a function. The limit of f(x) as x approaches a is L (written lim_(x→a) f(x) = L) if for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formal definition is the foundation for understanding continuity and other important concepts. The epsilon-delta definition is presented rigorously, emphasizing the precise meaning of the limit of a function. The definition includes the condition 0 < |x - a|, explicitly excluding the point x = a itself from the consideration of the limit. The use of ε and δ is standard notation in analysis, and this definition is fundamental for establishing a rigorous understanding of limits, which forms the basis for understanding continuity and other important analytical concepts.

Building on the definition of limits, the section defines continuity. A function f is continuous at a point a if lim_(x→a) f(x) = f(a). A function is continuous on a set if it's continuous at every point in that set. The section likely discusses properties of continuous functions, potentially mentioning the Extreme Value Theorem, which states that a continuous function on a closed, bounded interval attains both a maximum and a minimum value. This definition establishes a connection between the limit concept and the function's value at a point, formalizing the intuitive notion of a continuous function. The Extreme Value Theorem is a fundamental result for continuous functions on closed and bounded intervals, illustrating the significant properties associated with continuity. Exercises likely explore the properties of continuous functions on different types of intervals or sets, allowing for a deeper understanding of the implications of continuity.

The concept of limits is extended to include limits at infinity. The limit of f(x) as x approaches positive infinity is L (written lim_(x→+∞) f(x) = L) if for every ε > 0, there exists a real number M such that if x > M, then |f(x) - L| < ε. Limits as x approaches negative infinity are defined similarly. The section likely also introduces one-sided limits (limits from the right and from the left). These extensions of the limit concept broaden the scope of functions that can be analyzed. The definitions of limits at infinity and one-sided limits provide a more comprehensive treatment of limit behavior, enabling a deeper understanding of function behavior near infinity and at specific points. This expanded set of tools enhances the analytical power of the theory of limits and lays the groundwork for later analysis.

This subsection introduces Rolle's Theorem, a fundamental result in differential calculus. The theorem states that if a function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0. This theorem essentially says that if a differentiable function starts and ends at the same y-value, there must be at least one point where its derivative (slope) is zero—a horizontal tangent. The proof uses the Extreme Value Theorem, showing a connection between the properties of continuous functions (attaining maximum and minimum values) and the existence of critical points (where the derivative is zero). The geometrical interpretation of Rolle's Theorem is straightforward; it illustrates how the derivative provides information about the behavior of the function itself. Rolle's Theorem serves as a stepping stone towards the more general Mean Value Theorem.

The Mean Value Theorem is a generalization of Rolle's Theorem. It states that if a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a). This means there's a point where the instantaneous rate of change (the derivative) is equal to the average rate of change over the entire interval. The Mean Value Theorem is a cornerstone in calculus and has many applications. It's likely that the section provides examples showing how the Mean Value Theorem can be used to prove other results in calculus or analysis. Exercises might involve applying the Mean Value Theorem to solve problems related to the properties of differentiable functions or finding specific points where the derivative equals the average rate of change. The theorem provides a powerful connection between the average rate of change of a function and its instantaneous rate of change at some point within the interval.

This subsection likely presents applications of the Mean Value Theorem, potentially illustrating its use in proving other theorems or solving problems. For instance, the text might demonstrate how to use the Mean Value Theorem to show that a function is strictly monotonic or to prove the uniqueness of solutions to certain equations. It may also introduce higher-order derivatives, defining the second derivative, third derivative, and so on, and discussing their interpretations. The section's exercises likely reinforce the applications of the Mean Value Theorem in various contexts. The discussion of higher-order derivatives lays the groundwork for extending the concepts of differentiation to more complex functions and problems. This section emphasizes the practical significance of the Mean Value Theorem and extends the concepts of differentiation to include higher-order derivatives, increasing the applicability of differential calculus.

This subsection introduces the concept of higher-order derivatives. Given a function f that is differentiable on an open interval I, and if its derivative f' is also differentiable at a point a in I, then the derivative of f' at a is called the second derivative of f at a, denoted f''(a). This process can be continued to define higher-order derivatives (third derivative f'''(a), fourth derivative f''''(a), and so on). The nth derivative is denoted f⁽ⁿ⁾(a), where f⁽⁰⁾(a) represents the original function f(a). The existence of higher-order derivatives implies increasingly stronger smoothness conditions on the function. The notation for higher-order derivatives is clearly established, using the standard prime notation and superscripts. The definition is presented rigorously, outlining the conditions under which higher-order derivatives exist. The significance of the existence of higher-order derivatives is implicitly stated; it points to increased smoothness and regularity in the function's behavior.

This subsection introduces the Riemann integral, a method for defining integration. The section mentions that the definition provided is due to Darboux but is equivalent to the Riemann integral. The text highlights the distinction between the Riemann integral and other more general integral theories, such as the Lebesgue integral. The section also differentiates between the definite integral (a numerical value representing the area under a curve) and the indefinite integral (a family of antiderivatives). The terminology is carefully laid out, defining key concepts such as Riemann integrable functions, definite integral, and indefinite integral. The clarification that the Darboux definition is equivalent to the Riemann definition helps to contextualize this specific approach to integration within the broader framework of integral calculus. The distinction between definite and indefinite integrals is emphasized to avoid potential confusion, ensuring a clear understanding of the notation and interpretations used throughout the rest of the text.

This subsection introduces the sine and cosine functions, sin(x) and cos(x), respectively. The definitions are presented, likely based on a geometric or series representation, although the specific method isn't detailed in the provided text excerpt. The section states that the sine and cosine functions are continuous on the entire real line (R). This basic property of continuity is stated without proof, likely relying on prior knowledge or a preceding section's results. The functions' properties, including continuity, are foundational for later discussions of their derivatives and other analytical properties. The fact that continuity is explicitly stated highlights its significance for understanding the behavior of these functions. The brief introduction lays the groundwork for a more in-depth analysis of trigonometric functions' properties and applications within the framework of real analysis.

The tangent function, tan(x), is defined (likely using the relationship to sine and cosine), and its properties are presented. The section states that the tangent function has a domain D (a specific subset of R), a range of all real numbers (R), and is differentiable at every point in its domain. Moreover, it's stated that the tangent function is strictly increasing on each interval in its domain. The introduction of the tangent function extends the discussion beyond sine and cosine, expanding the scope to other important trigonometric functions. The specification of the domain and range is important for understanding the function's behavior. The statement regarding differentiability and monotonicity provides crucial information for further analysis, particularly when studying derivatives and related concepts. The concept of periodicity for functions is also defined, although its immediate application to trigonometric functions might be in a following section.

This subsection defines periodicity of a function and likely applies this definition to the trigonometric functions. A function f is periodic if there exists a real number p > 0 such that f(x + p) = f(x) for all x in the domain. The smallest such positive p is called the period of f. The section may include the discussion of trigonometric identities, such as the addition formula for cosine. For example, it might show that cos(x + y) = cos(x)cos(y) - sin(x)sin(y). This demonstration of trigonometric identities is relevant to the analytical manipulation of trigonometric functions in various contexts, expanding the tools available for their analysis. The discussion of periodicity is foundational for understanding the repetitive behavior of trigonometric functions, which is a key characteristic. The inclusion of trigonometric identities allows for a more comprehensive treatment of these functions within the framework of real analysis.