Introduction to Real Analysis

1. Real Analysis: The Foundation for Advanced Math

The study of real analysis is indispensable for a prospective graduate student of pure or applied mathematics.

Beyond Formulas. Real analysis transcends routine formula manipulation, fostering deductive thinking and analytical skills applicable across mathematical disciplines and fields like economics and computer science. It's about understanding why things work, not just how.

Rigorous Proofs. A core element of real analysis is the construction and comprehension of mathematical proofs. This involves mastering precise definitions, logical arguments, and the ability to extend ideas to new contexts. The journey from initial mystery to comfortable understanding is a rewarding, albeit challenging, process.

Essential for Graduate Studies. Real analysis provides the bedrock for advanced mathematical study. It equips students with the tools to rigorously examine mathematical situations and extend concepts to novel scenarios, making it invaluable for graduate-level work in pure and applied mathematics.

2. Understanding the Real Number System

The real number system can be described as a "complete ordered field," and we will discuss that description in considerable detail.

Algebraic and Order Properties. The real number system (ℝ) possesses fundamental algebraic properties (field axioms) governing addition and multiplication, along with order properties defining positivity and inequalities. These properties form the basis for algebraic manipulations and working with inequalities.

Completeness Property. The completeness property, specifically the Supremum Property, distinguishes ℝ from the rational numbers (ℚ). This property ensures that every nonempty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ, a crucial foundation for limit theory.

Rational vs. Irrational. While rational numbers (ℚ) are dense in ℝ, meaning there's a rational number between any two real numbers, irrational numbers also exist and are, in a sense, "more numerous." The uncountability of ℝ demonstrates a deeper level of infinity than that of ℚ.

3. Sequences and Their Limits: The Building Blocks of Analysis

In this chapter, we will be concerned with sequences in lR and will discuss what we mean by the convergence of these sequences.

Defining Convergence. A sequence (xn) converges to x if, for any desired level of closeness (ε), there exists a point in the sequence (K) beyond which all terms are within ε of x. This precise definition is crucial for rigorous analysis.

Limit Theorems. Limit theorems provide tools for calculating limits of sequences formed by algebraic operations on convergent sequences. These theorems simplify the process of finding limits and are essential for more advanced analysis.

Cauchy Sequences. Cauchy sequences are sequences whose terms become arbitrarily close to each other. The Cauchy Criterion states that a sequence converges if and only if it is a Cauchy sequence, providing a way to determine convergence without knowing the limit in advance.

4. Mastering Limits: Approaching the Infinitesimal

If the successive values attributed to the same variable approach indefinitely a fixed value, such that they finally differ from it by as little as one wishes, this latter is called the limit of all the others.

Formal Definition. The concept of a limit captures the behavior of a function as its input approaches a specific value. The formal ε-δ definition provides a rigorous way to express this idea, ensuring that the function's output can be made arbitrarily close to the limit by choosing inputs sufficiently close to the target value.

Sequential Criterion. The sequential criterion provides an alternative way to characterize limits using sequences. It states that a function has a limit at a point if and only if for every sequence approaching that point, the corresponding sequence of function values converges to the same limit.

Limit Theorems. Limit theorems allow us to calculate limits of combinations of functions based on the limits of the individual functions. These theorems simplify the process of finding limits and are essential for more advanced analysis.

5. Continuous Functions: The Backbone of Real Analysis

The function f is said to be continuous at c if, given any number 8 > 0, there exists 8 > 0 such that if x is any point of A satisfying lx - cl < 8, then lf(x) -f(c) l < 8.

Definition of Continuity. A function is continuous at a point if small changes in the input result in small changes in the output. The formal ε-δ definition captures this idea precisely, ensuring that the function's output can be made arbitrarily close to its value at the point by choosing inputs sufficiently close to the point.

Sequential Criterion. The sequential criterion provides an alternative way to characterize continuity using sequences. It states that a function is continuous at a point if and only if it preserves convergence of sequences; that is, for every sequence converging to the point, the corresponding sequence of function values converges to the function's value at the point.

Uniform Continuity. Uniform continuity is a stronger condition than pointwise continuity. It requires that the same δ "works" for all points in the domain, ensuring that the function's behavior is "uniformly smooth" across the entire set.

6. Differentiation: Unveiling Rates of Change

If f : I ----) lR has a derivative at c E I, then f is continuous at c.

Definition of the Derivative. The derivative of a function at a point measures the instantaneous rate of change of the function at that point. It is defined as the limit of the difference quotient as the change in input approaches zero.

Differentiation Rules. Differentiation rules provide formulas for calculating derivatives of combinations of functions, such as sums, differences, products, quotients, and compositions. These rules simplify the process of finding derivatives and are essential for more advanced analysis.

Mean Value Theorem. The Mean Value Theorem relates the values of a function to the values of its derivative. It states that for a differentiable function on an interval, there exists a point where the tangent line is parallel to the secant line connecting the endpoints of the interval.

7. The Riemann Integral: Summing the Infinitesimal

This approach has the advantage that it is consistent with the students' first exposure to the integral in calculus, and since it is not dependent on order properties, it permits immediate generalization to complex- and vector-values functions that students may encounter in later courses.

Riemann Sums. The Riemann integral is defined as the limit of Riemann sums, which are approximations of the area under a curve using rectangles. The integral exists if these sums converge to a unique value as the width of the rectangles approaches zero.

Integrability Conditions. Not all functions are Riemann integrable. Continuous and monotone functions on closed bounded intervals are Riemann integrable, but discontinuous functions may not be. The Lebesgue Integrability Criterion provides a definitive characterization of Riemann integrable functions.

Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. It has two parts: (1) the derivative of the indefinite integral of a function is the function itself, and (2) the definite integral of a function can be evaluated by finding an antiderivative and evaluating it at the endpoints of the interval.

8. Sequences of Functions: Convergence and Interchange of Limits

Sequences of functions and uniform convergence are discussed in the first two sections of Chapter 8, and the basic transcendental functions are put on a firm foundation in Sections 8.3 and 8.4.

Pointwise vs. Uniform Convergence. A sequence of functions can converge pointwise, meaning that for each x in the domain, the sequence of function values converges. However, this does not guarantee that the limit function will inherit properties like continuity or differentiability. Uniform convergence, a stronger condition, ensures that the entire sequence of functions converges "evenly" across the domain.

Uniform Convergence and Continuity. A uniformly convergent sequence of continuous functions has a continuous limit. This is a crucial result, as it allows us to interchange the limit operation with the evaluation of continuity.

Applications to Transcendental Functions. Uniform convergence is used to rigorously define and establish the properties of transcendental functions like exponentials, logarithms, and trigonometric functions. This provides a solid analytical foundation for these essential functions.

9. Infinite Series: Summing to Infinity

Chapters 8 and 9 are intrinsically important, and they also show how the material in the earlier chapters can be applied.

Absolute vs. Conditional Convergence. A series is absolutely convergent if the sum of the absolute values of its terms converges. Absolute convergence implies convergence, but the converse is not always true. Conditionally convergent series converge, but not absolutely.

Convergence Tests. Various tests, such as the Ratio Test, Root Test, and Integral Test, are used to determine the convergence or divergence of infinite series. These tests provide practical tools for analyzing the behavior of series.

Series of Functions. The concepts of pointwise and uniform convergence extend to series of functions. Uniform convergence of a series of functions ensures that the limit function inherits properties like continuity and integrability from the individual terms.

10. Topology: Abstracting the Concepts of Openness and Closeness

Earlier theorems and proofs are extended to a more abstract setting.

Open and Closed Sets. Open sets are sets where every point has a neighborhood contained within the set, while closed sets contain all their limit points. These concepts generalize the idea of open and closed intervals on the real line.

Compact Sets. Compact sets are sets where every open cover has a finite subcover. In the real line, compact sets are precisely the closed and bounded sets, as stated by the Heine-Borel Theorem.

Continuous Functions on Compact Sets. Continuous functions on compact sets have special properties, such as attaining maximum and minimum values and being uniformly continuous. These properties are essential for many applications in analysis.

Last updated: March 15, 2025