Real Analysis: Sequences & Series

The section initiates the study of real analysis by focusing on the real number system (R), highlighting its completeness property as the cornerstone for subsequent concepts. This property ensures that every non-empty set of real numbers bounded above has a least upper bound (supremum), and similarly, every non-empty set bounded below has a greatest lower bound (infimum). This foundational concept is crucial for establishing theorems about limits and sequences of real numbers. The text emphasizes that this least-upper-bound property is more crucial for analysis than simply considering R as an ordered field. The absence of this property in the rational numbers (Q) is illustrated by the example of the square root of two, demonstrating the existence of real numbers that are not rational. The concept of supremum and infimum are used frequently throughout the text, and the difference between supremum/infimum and maximum/minimum, where the latter requires the supremum/infimum to be in the set, is clearly defined. The importance of understanding this fundamental property is consistently reinforced throughout the text to build a strong foundation for the concepts to come.

The text introduces fundamental concepts of naïve set theory, which forms the language of modern analysis. It emphasizes the importance of understanding sets, while assuring readers that the approach is informal and widely used. A quick refresher is provided for those familiar with the basics. The section explores the distinction between countable and uncountable sets. The countability of the rational numbers (Q) is demonstrated through an informal counting argument, while Cantor's diagonal argument is used to prove the uncountability of the real numbers (R). The text highlights a key result: If a set A is a subset of a countable set B, then A is also countable. The contrapositive of this statement – if A is uncountable, then B is uncountable – is presented as a useful tool. The concept of ordered sets is also introduced, explaining how sets like rational, integer, and natural numbers are ordered, and presenting other examples such as the lexicographic order of words (as used in dictionaries) to broaden the student's understanding of order beyond purely numerical contexts. This understanding of set theory provides the essential framework for constructing and manipulating mathematical objects throughout the real analysis course.

This subsection builds upon the introduction of sets and moves towards formalizing the properties of real numbers. The concept of an ordered field is introduced; a field that has an order relation satisfying specific axioms. A proposition is presented and proven showing that in an ordered field, if the product of two elements is positive, both must be positive or both must be negative. The section then delves deeper into the properties of the real numbers (R), emphasizing their unique role within the framework of ordered fields. It contrasts the properties of the real numbers with the rational numbers, underscoring the significance of the least-upper-bound property of the real numbers. The lack of this property in the rational numbers is highlighted. The section explores the implications of the least-upper-bound property, explaining how this property implies the existence of real numbers which are not rational, such as the square root of two. The significance of the least-upper-bound property in analysis is stressed, establishing this property as a foundational building block of real analysis. The concepts explored here are crucial for a solid grasp of subsequent sections concerning limits, continuity, and integration.

This section introduces the fundamental concept of convergence for sequences. The definition of a convergent sequence is rigorously presented, emphasizing the intuitive notion that the terms of the sequence eventually get arbitrarily close to the limit. This is contrasted with the idea that the sequence may never actually reach the limit. The section includes examples illustrating the concept, clarifying the difference between a sequence that converges to a limit and one that does not. The importance of understanding this intuitive aspect alongside the formal definition is highlighted. A key idea presented is that while the sequence terms eventually become close to the limit, there's no requirement that any term in the sequence equals the limit. This careful explanation of the concept of convergence lays a strong foundation for the subsequent discussions of limits of functions and other advanced topics. Visual aids, such as graphs, are mentioned as ways to understand the concept of sequences and their limits. Newton's method for approximating the square root of 2 is used as an illustrative example of a sequence that converges rapidly to a limit.

The focus shifts to monotone sequences – those that are either entirely non-decreasing or non-increasing. The section explains that a monotone sequence converges if and only if it is bounded. The Monotone Convergence Theorem is presented as a formal statement of this idea, providing a powerful tool for determining whether or not a monotone sequence converges. The importance of establishing boundedness as a necessary condition for convergence is emphasized through examples. A cautionary example is given highlighting a monotone increasing sequence (related to the harmonic series) that is unbounded and therefore divergent, reinforcing the crucial role of boundedness in the context of the monotone convergence theorem. Examples of sequences, including those that are bounded and unbounded, are provided to reinforce understanding and provide context. The text explicitly states that a proof showing that the monotone sequence is bounded is necessary to use the monotone convergence theorem to conclude that a sequence converges. This section shows the practical applications of theoretical ideas in a clear and concise manner.

This subsection introduces the concepts of limit inferior (liminf) and limit superior (limsup) which are particularly useful for analyzing the behavior of sequences that might not converge. It's explained that these concepts can be applied to any bounded sequence, allowing a more complete analysis even when standard limits don't exist. The advantages of liminf and limsup are detailed—they provide a way to extract information about the ultimate behavior of a sequence even if it doesn't converge to a single limit. The text notes that while working with liminf and limsup is similar to working with limits, there are subtle differences that require careful attention. The section touches upon how these concepts could be applied to unbounded sequences using extended real numbers (R*), but notes that R* is not a field, and so this extended application is relegated to exercises. A connection is made to the case where a sequence converges, showing that if the limit inferior and limit superior are equal, then the sequence converges. This section offers more advanced tools for analyzing sequences.

The section introduces Cauchy sequences—sequences where the terms get arbitrarily close to each other as the sequence progresses. A key result is presented: Every Cauchy sequence in the real numbers converges. This property is often used as an alternative definition of the completeness of the real numbers. The text explains that the real numbers are Cauchy-complete because they possess the least-upper-bound property, a connection that reinforces the fundamental importance of the least-upper-bound property discussed earlier. The section also mentions the possibility of constructing the real numbers by adding enough points to the rational numbers to ensure all Cauchy sequences converge; however, details of this construction are omitted. The discussion underscores the profound connection between Cauchy sequences and the completeness property, offering a different perspective on the fundamental nature of the real number system and its role in the study of real analysis. The concept of Cauchy sequences plays an important role in more advanced mathematical analysis.

This section turns its attention to infinite series—the sum of infinitely many terms of a sequence. It notes the added complexity of multiplying series compared to adding them. The focus then shifts to absolute convergence, explaining that series with non-negative terms are easier to analyze because their partial sums form a monotone increasing sequence. The section presents the p-series test as a crucial tool for determining the convergence or divergence of p-series, ∑(1/n^p), where p is a positive number, demonstrating how to apply comparison tests. It's noted that these convergence tests don't indicate the exact value of the sum (only whether it exists), providing an example of a convergent series where finding the sum is challenging. The concept of the Riemann zeta function is mentioned as a related, fascinating example from number theory, showcasing the broad connections between different areas of mathematics. The limitations of simple comparison methods and the deeper mathematical significance of convergence tests are discussed.

This section presents a rigorous definition of continuity for functions, moving beyond the intuitive notion of being able to draw a graph without lifting a pen. The formal definition using the epsilon-delta (ε-δ) criterion is introduced, emphasizing its importance for precise mathematical reasoning. The text contrasts this rigorous approach with the more informal understanding often encountered in high school calculus. The historical development of the definition, involving contributions from mathematicians like Bolzano, Cauchy, and Weierstrass, is briefly mentioned, highlighting the evolution of this critical concept. The importance of the rigorous definition is repeatedly stressed, setting the stage for more advanced concepts and theorems related to continuity. The contrast between the intuitive understanding of continuity and the need for rigorous mathematical treatment establishes the need for a deeper understanding of the foundational concepts of real analysis.

The concept of uniform continuity is introduced, distinguishing it from simple continuity. The key difference lies in the dependence of the choice of δ (in the ε-δ definition) on the point c in the domain for continuity, and the independence of δ from c in uniform continuity. The text explains that in uniform continuity, a single δ value works for all points within the domain, for a given ε. This nuance highlights how the size and properties of the domain impact the continuity of a function. The section notes that every uniformly continuous function is also continuous, and that the domain of definition is crucial in determining whether a function is uniformly continuous. An example is given to demonstrate that a function that fails to be uniformly continuous on a larger set might still be uniformly continuous when restricted to a smaller subset. This emphasizes the importance of considering both the function's properties and its domain of definition when working with uniform continuity.

The Intermediate Value Theorem is presented as a consequence of the continuity of a function. The text states that if a function is continuous on a closed interval, it takes on all values between its values at the endpoints of the interval. This theorem is shown to have practical applications in finding roots of equations. The bisection method is highlighted as a simple, yet effective algorithm that leverages the Intermediate Value Theorem to approximate roots of continuous functions to any desired accuracy in a finite number of steps. The discussion emphasizes that this method works for all continuous functions, not just polynomials, further reinforcing the importance and versatility of the Intermediate Value Theorem. The simplicity of the bisection method contrasted with other more advanced root finding methods further highlights its importance in providing a guaranteed method to find roots of a function.

The section introduces differentiation, defining critical points as points where the derivative is zero or undefined. A key result is mentioned – a relative minimum or maximum at an interior point of an interval must be a critical point. This provides a strategy for finding extrema: locating critical points and comparing function values at critical points and endpoints. The Mean Value Theorem is introduced, explaining that for a differentiable function on a closed interval, there exists at least one point where the instantaneous rate of change equals the average rate of change over the interval. The practical applications of the Mean Value Theorem are discussed and an interesting note about its use in legal contexts (speed measurement) is shared to highlight the theorem’s relevance outside the realm of pure mathematics. The discussion of critical points and their relationship to relative extrema demonstrates the importance of differentiation in optimization problems and finding extreme values.

This section introduces Taylor's Theorem, which provides a way to approximate a function using polynomials. It generalizes the idea from the Mean Value Theorem, where a function can be approximated near a point using its first derivative. Taylor's Theorem extends this approximation to use higher-order derivatives. The text explains that the error in the approximation depends on the higher-order derivatives and the distance from the point of approximation. The accuracy of this approximation is discussed, noting that it may not be uniformly good across the entire domain. An example is given where Taylor polynomials provide increasingly good approximations for some values, but actually provide worse approximations for other values, highlighting the limitations of Taylor approximations and showing that more advanced techniques may be needed for other domains. The concept of Taylor series, viewed as an infinite sum of polynomials, is implicitly introduced through this discussion.

This section introduces the Riemann integral, a fundamental concept in calculus and analysis. The text clarifies a common point of confusion among students—the difference between the Riemann integral and the antiderivative. The Riemann integral is defined informally as the area under a curve, emphasizing that this is distinct from the antiderivative. This section utilizes Darboux sums to define the Riemann integral, presenting a technically simpler (but equivalent) approach compared to Riemann's original definition, which uses tagged partitions. The choice of Darboux sums is explicitly justified as being technically simpler. The section addresses a common misconception among calculus students: that integrals without closed-form solutions are somehow flawed. The text clarifies that most integrals do not have closed-form solutions and that even seemingly closed-form integrals often rely on computational approximations. An example using the natural logarithm (ln x), which is defined as the antiderivative of 1/x, is provided to highlight this point. The use of Darboux sums provides a structured and rigorous foundation for understanding the Riemann integral, making it accessible and less prone to misinterpretations.

Building on the definition of the Riemann integral, this section explores its key properties. While the specific details of these properties are not given in the summary, it is mentioned that the section includes a proof of the Fundamental Theorem of Calculus, which establishes the crucial link between integration and differentiation. This theorem is a cornerstone result in calculus, demonstrating the inverse relationship between integration and differentiation. It explains how to compute definite integrals using antiderivatives, providing a powerful tool for evaluating definite integrals. The section also touches upon more advanced topics, including sequences of functions and the interchange of limits, indicating that these concepts will be explored in detail later in the course. The emphasis on the Fundamental Theorem and the brief mention of other advanced topics highlight the scope and depth of the material covered in this section and demonstrate the significance of the Riemann integral in the overall structure of real analysis.

This section introduces improper integrals, extending the concept of the Riemann integral to handle cases where the interval of integration is unbounded (e.g., integrating over the entire real line or an unbounded interval) or the function being integrated is unbounded on a bounded interval. The text makes it clear that such functions are not Riemann integrable in the standard sense. The definition of an improper integral is given as a limit of integrals, which handles these cases where standard Riemann integration fails. The section provides examples of how to evaluate improper integrals and notes when such integrals converge or diverge. The cautionary note about double limits emphasizes the potential for complications when dealing with improper integrals defined as limits involving two parameters. It highlights the need for careful analysis and rigorous justification when using improper integrals. The extension of integration to these more challenging cases reveals the versatility and power of the Riemann integral in handling a broad range of functions and domains, showing how to tackle integration problems that wouldn't be directly accessible using standard Riemann integration.

This section provides context by describing how the text originated as lecture notes for a real analysis course (Math 444) at the University of Illinois at Urbana-Champaign (UIUC) in Fall 2009. Subsequent additions, including a chapter on metric spaces (for Math 521 at the University of Wisconsin–Madison, UW), and material for a year-long course (Math 4143/4153 at Oklahoma State University, OSU), are detailed. The text is compared to Bartle and Sherbert's Introduction to Real Analysis, a standard textbook for UIUC Math 444, noting similarities in the beginning structure and a key difference in the definition of the Riemann integral (using Darboux sums instead of tagged partitions). The text also mentions how the first few chapters could be used in an introductory proofs course at Iowa State University (Math 201), often in conjunction with Hammack's Book of Proof. This section provides a clear picture of the target audience, the course context, and the relationship to other well-established texts in the field of real analysis. The section also emphasizes that a basic proof course is a prerequisite for the courses this textbook is designed for.

The author mentions several other real analysis textbooks, highlighting Rudin's Principles of Mathematical Analysis ('baby Rudin') as a source of inspiration, while acknowledging that Rudin's text is more advanced. Rosenlicht's Introduction to Analysis is presented as a less expensive, simpler alternative, and William Trench's freely downloadable Introduction to Real Analysis is also mentioned. This section demonstrates the author's awareness of the landscape of real analysis textbooks and provides suggestions for further reading. The author also discusses their preferred proof style—favoring direct proofs or contrapositive arguments over proofs by contradiction, except when the contrapositive is awkward, or contradiction is quick. This insight into the author's pedagogical approach provides valuable context for understanding the style and presentation of the material in this text. The choices made highlight potential benefits for students who might find proofs by contradiction more challenging. Providing these alternative resources is useful for students looking for supplementary materials.

This section clearly states the open-access licensing of the textbook, specifying the Creative Commons license (either CC-BY-NC-SA or CC-BY-SA). The permissive nature of the license is explicitly stated: users can use, print, duplicate, and share the book freely, with conditions that derivative works must use at least one of the specified licenses and be clearly marked as such. This section establishes that the text is an open educational resource (OER), emphasizing its accessibility and promoting its use for educational purposes in various contexts. The clear articulation of licensing terms assures users of their rights and responsibilities concerning the book and related materials. Providing the Creative Commons contact information allows readers to further clarify any licensing questions or concerns, ensuring clarity and facilitating widespread use of the material.