Mathematical Analysis I

This subsection introduces fundamental concepts of set theory, essential for building a strong foundation in real analysis. A set is defined intuitively as a collection of objects with specific properties; these objects are termed elements or members. Sets are typically denoted using uppercase letters, while elements are represented by lowercase letters. Set membership is indicated using the ∈ symbol (e.g., a ∈ A signifies that 'a' is an element of set 'A'), and non-membership is denoted by ∉. The section explains two primary methods for specifying a set: listing all its elements (if feasible) or using a defining rule. For example, a set A containing elements a, b, c, and d is represented as A = {a, b, c, d}. This foundational understanding of sets forms the basis for subsequent concepts and proofs within the lecture notes. The intuitive nature of the explanation makes it suitable for students with diverse mathematical backgrounds, potentially easing their transition into more abstract real analysis concepts.

This subsection focuses on essential tools within mathematical analysis. It begins by defining a function as a collection of ordered pairs, aligning with the graphical representation of functions in calculus. The equivalence of referring to a function 'f' and its graph {(x, f(x)) : x ∈ X} is highlighted. The equality of two functions is established as the equivalence of their corresponding subsets of X × Y. This section dives into critical function properties, defining a function as surjective (or onto) if every element in the codomain has a pre-image in the domain, and injective (or one-to-one) if distinct elements in the domain map to distinct elements in the codomain. A significant portion is dedicated to the principle of mathematical induction, presented as a method to prove a sequence of propositions P(n), for each natural number n. The section details the base case (P(1) is true) and the inductive step (if P(k) is true, then P(k+1) is true). This principle, also referred to as strong induction, becomes indispensable for many subsequent proofs within the lecture notes. The strong induction principle, which allows assuming the truth of all preceding propositions in the inductive step, is introduced and proven equivalent to standard induction.

Building upon the earlier introduction of sets and functions, this subsection delves into additional concepts related to infinite sets. A key result demonstrates that for any infinite set A, a one-to-one function from the set of natural numbers (N) to A exists. This theorem forms a basis for future discussions on sequences and accumulation points. The section introduces a proof by contradiction and further expands the properties of functions and sets. The proof rigorously demonstrates the existence of a one-to-one mapping from natural numbers to an infinite set, a result highly relevant for subsequent exploration of sequences and their convergence. Additionally, an example illustrating proof by induction is provided, showing that every positive integer greater than 1 can be expressed as either a prime number or a product of prime numbers. This example showcases the practical application of mathematical induction, a crucial proof technique. This reinforces the importance of mathematical induction as a fundamental tool for proving statements involving natural numbers and strengthens the reader’s understanding.

This section establishes the axiomatic foundation for real numbers by defining them as an ordered field. The text introduces thirteen axioms that govern the operations of addition (+) and multiplication (⋅) and the order relation (<) within the set of real numbers (R). These axioms define the fundamental properties of real numbers, including commutativity, associativity, distributivity, the existence of additive and multiplicative inverses (except for zero), and the properties of the order relation. The section highlights that both the real numbers (R) and the rational numbers (Q) satisfy these axioms, demonstrating that they are both examples of ordered fields. However, the integers (Z) and natural numbers (N) are explicitly stated as not forming fields because they lack multiplicative inverses for all elements except 1 and -1. This sets the stage for the introduction of the completeness axiom, which further distinguishes the real numbers from other ordered fields. The inclusion of the natural numbers (N), integers (Z), and rational numbers (Q) as subsets within the real numbers (R) is also briefly explained, demonstrating how the real number system encompasses these number systems. This forms a crucial bridge between familiar number systems and the more abstract concept of an ordered field, making the transition smoother for students.

This crucial section introduces the completeness axiom, the property that distinguishes the real numbers (R) from all other ordered fields. Before stating the axiom, the text clarifies that many ordered fields exist, emphasizing the uniqueness of the real number system. The completeness axiom is then presented: every nonempty subset A of R that is bounded above has a least upper bound, also known as the supremum (sup A). This supremum is a real number, and its existence is guaranteed by the axiom. The section underscores the pivotal role of the completeness axiom in the proofs of fundamental theorems within real analysis, highlighting its importance in distinguishing real numbers from other ordered fields. The geometric interpretation of the absolute value |a| as the distance from a number 'a' to 0 is also provided, extending the concept to the distance between two points, d(a,b) = |a - b|. The section includes a proof demonstrating that if α and β both satisfy the conditions of being the supremum of a set A, then α must equal β, demonstrating the uniqueness of the supremum. This emphasizes the precision and consistency inherent in the definition and properties of the real numbers defined by these axioms.

This section introduces the concept of sequences, defining them as functions with the natural numbers (N) as their domain. The notation for sequences, such as {aₙ}∞ₙ=₁, {aₙ}ₙ, or {aₙ}, is established, where each aₙ represents the nth term of the sequence. The section then proceeds to define the limit of a sequence. A sequence {aₙ} converges to a limit 'a' if, for any given positive epsilon (ε), there exists a natural number N such that for all n > N, the absolute difference |aₙ - a| is less than ε. This formal definition establishes the criteria for sequence convergence. The concept of a divergent sequence, one that does not converge, is also introduced. A key remark highlights that, using the Archimedean property, a sequence {aₙ} converges to 'a' if and only if for any ε > 0, there exists a real number N such that for all n > N, |aₙ - a| < ε. This provides an alternative, equivalent characterization of sequence convergence, demonstrating the rigorous approach of the lecture notes. The section sets the stage for a deeper exploration of sequence properties and convergence behaviors in subsequent sections.

Building on the concept of convergent sequences, this subsection delves into the existence of convergent subsequences within bounded sequences. It's proven that every bounded sequence contains a convergent subsequence. For a finite set of sequence values, at least one value must appear infinitely often, forming a constant (and thus convergent) subsequence. For infinite sets of values, the proof constructs a sequence of nested closed bounded intervals. These intervals are created by repeatedly bisecting intervals, ensuring that at least one resulting interval contains infinitely many sequence terms. The nested interval theorem guarantees the existence of a point within all these intervals, forming the limit of the convergent subsequence. This is then linked to the Bolzano-Weierstrass theorem, which formally states that every bounded sequence in R has a convergent subsequence. This theorem provides a crucial tool for establishing convergence within more complex sequences and lays the foundation for further analysis of sequence properties. The rigorous proof method emphasizes the mathematical precision of the concepts and provides a solid basis for future applications.

This section introduces the concept of compact sets in the context of sequences and their convergence. A subset A of R is defined as compact if every sequence in A possesses a subsequence that converges to a point within A. This definition uses sequential compactness, which is presented as more intuitive for students than the general definition involving open covers. The section then proves that a closed bounded interval [a, b] is compact, leveraging the Bolzano-Weierstrass theorem to establish the existence of a convergent subsequence and demonstrating that the limit of this subsequence falls within the interval [a, b]. The section concludes with the definition of a limit point (also called cluster point or accumulation point). A point 'a' is a limit point of a set A if every open interval around 'a' contains infinitely many points of A, extending the concept of limits to sets and reinforcing the fundamental idea of a limit and its relationship to convergence within the broader field of real analysis. The connection between compact sets and the existence of convergent subsequences within them is crucial for later concepts and theorems.

This section formally defines continuity of a function. The definition emphasizes that a function f is continuous at a point x₀ in its domain D if, for any given positive epsilon (ε), there exists a positive delta (δ) such that for all x in D satisfying |x - x₀| < δ, the inequality |f(x) - f(x₀)| < ε holds. This definition is carefully constructed to encompass points x₀ that are not limit points of D. The text explicitly notes that if x₀ is an isolated point of D, then every function is continuous at x₀ because there exists a δ such that the interval (x₀ - δ, x₀ + δ) contains only x₀ itself, making the condition |f(x) - f(x₀)| < ε trivially true. This refined definition of continuity is crucial because it handles isolated points within the domain, ensuring generality and avoiding ambiguity in the definition. The section subtly lays the groundwork for understanding that continuity is a local property of a function, relying on the behavior of the function in an arbitrarily small neighborhood around a specific point.

This subsection introduces the Intermediate Value Theorem (IVT), a fundamental result in real analysis. The IVT states that if f is a continuous function on a closed interval [a, b], and γ is any value between f(a) and f(b), then there exists at least one c in the interval (a, b) such that f(c) = γ. This theorem is then applied to prove several important results. One example demonstrates that every positive real number has a positive nth root, using the IVT and properties of the function f(x) = xⁿ. Another example uses the IVT to show the existence of a real solution to the equation eˣ = -x. This showcases the power of the IVT in proving the existence of solutions to equations without explicitly finding the solution itself. The section also includes a corollary stating that for a continuous function f on [a, b], with m and M as the minimum and maximum values of f on the interval, respectively, for every γ in the range [m, M] there exists a c in [a, b] such that f(c) = γ. This is a direct consequence of the Intermediate Value Theorem and highlights its significance in characterizing continuous functions.

This section introduces the concept of uniform continuity, a stronger condition than pointwise continuity. The text observes that if a function is uniformly continuous on a set D, it's also uniformly continuous on any subset of D. A function f is uniformly continuous on D if, for any ε > 0, there exists a δ > 0 such that for all u and v in D, if |u − v| < δ, then |f(u) − f(v)| < ε. The key difference from standard continuity lies in the fact that δ depends only on ε and not on the specific point in D. This subtlety distinguishes uniform continuity, making it a stronger condition with significant implications in analysis. The section provides a result characterizing uniform continuity on open bounded intervals. It is noted that if a function is not uniformly continuous, there must exist sequences {xₙ} and {yₙ} such that |xₙ - yₙ| approaches 0, while |f(xₙ) - f(yₙ)| does not. This characterization provides a method for determining whether a function is uniformly continuous. The detailed explanation focuses on the rigorous differences between pointwise and uniform continuity.

This section formally defines the derivative of a function. A function f: G → R, where G is an open set, is differentiable at a point x ∈ G if the limit lim┬(h→0)⁡〖(f(x + h) - f(x))/h〗 exists. This limit, if it exists, is denoted as f'(x) or f⁽¹⁾(x) and represents the derivative of f at x. Higher-order derivatives are defined recursively: a function is twice differentiable if its first derivative is also differentiable, and the second derivative is obtained by differentiating the first derivative. This recursive process extends to define nth-order derivatives. The section establishes basic properties of derivatives, laying the groundwork for later applications. The concept of differentiability implies continuity, although the converse isn't necessarily true. The section also mentions Fermat's Rule, a method for identifying local extrema of a function by examining its derivative, linking the concept of the derivative to optimization problems. Pierre de Fermat's contribution to finding maxima and minima using derivatives is briefly mentioned, historically situating the derivative's importance in optimization problems.

This section centers on the Mean Value Theorem (MVT), a cornerstone of calculus. The MVT is presented as a consequence of Rolle's Theorem, a simpler version stating that if a function is continuous on a closed interval [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0. The MVT is then stated: if f is continuous on [a, b] and differentiable on (a, b), there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a). The theorem is interpreted geometrically as the existence of a tangent line parallel to the secant line connecting the endpoints of the curve. The section emphasizes the importance of the MVT as a powerful tool in calculus, frequently used to prove other important results and used to demonstrate the differentiability of an inverse function and find its derivative. Augustin-Louis Cauchy’s contribution in formalizing this theorem is referenced, highlighting its historical significance within mathematical analysis.

This subsection introduces L'Hôpital's Rule, a method for evaluating limits of indeterminate forms (0/0 or ∞/∞). The rule states that if lim┬(x→x₀)⁡〖f(x) = 0〗 and lim┬(x→x₀)⁡〖g(x) = 0〗, and lim┬(x→x₀)⁡〖f'(x)/g'(x)〗 exists, then lim┬(x→x₀)⁡〖f(x)/g(x) = lim┬(x→x₀)⁡〖f'(x)/g'(x)〗. The applicability of L'Hôpital's Rule is demonstrated through an example where the limit is evaluated by applying the rule, emphasizing that the rule requires conditions such as the continuity and differentiability of functions on a specific interval around the limit point, and that g'(x) should not be zero in that interval. L’Hôpital’s Rule efficiently resolves limits which are otherwise difficult to evaluate directly. The conditions for applying this rule are clearly specified to highlight the importance of careful application to ensure accuracy.