Basic Analysis: Introduction to Real Analysis, Volume I

About this book

Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl develops the proofs behind familiar ideas from calculus. The linked author-hosted PDF is the sixth edition, version 6.3, dated May 15, 2026, and contains an introduction followed by seven chapters. This guide follows that file rather than confusing it with the separate multivariable Volume II. Learners need familiarity with basic proof methods, sets, functions, and quantifiers. The book progresses from completeness of the real numbers through sequences, continuity, differentiation, integration, convergence of functions, and metric spaces. Its recurring challenge is to distinguish an intuitive picture or numerical pattern from a statement whose hypotheses and conclusion have been proved.

Chapter-by-chapter study guide

Introduction: Analysis, Sets, and Proof Language

The introduction explains the role of analysis and reviews the set-theoretic language needed for later arguments. Sets, functions, relations, cardinality, and logical structure make definitions precise. A proof must connect the stated assumptions with the actual conclusion; examples can illustrate an idea or disprove a universal claim, but a collection of favorable examples does not establish it. Quantifier order is especially important in definitions involving bounds and limits. The author also describes different course paths through the volume, allowing a slower real-line course and a faster route to metric spaces. The introduction should be used to identify gaps in proof fluency before abstract definitions accumulate.

Study application: Translate a statement about all inputs into explicit quantifiers and write its negation. Give one example, one nonexample, and a proof plan for a simple set or function claim, explaining the different role of each.

Chapter 1: Real Numbers

The chapter builds analysis around ordered-field properties and completeness rather than treating the number line only as a picture. Upper and lower bounds differ from maxima and minima, while a supremum need not belong to the set it bounds. Completeness supports later existence arguments and distinguishes the real line from familiar incomplete subsets. Absolute value expresses distance and supports inequalities central to limit proofs. Intervals, the size of the real numbers, and decimal representations connect structural properties with ordinary numerical notation. A useful habit is separating the property used in each step: an algebraic manipulation, an order argument, and an appeal to completeness are different contributions. Definitions should be tested on sets with and without attained endpoints.

Study application: Find bounds, supremum, infimum, maximum, and minimum for several sets, including one with an excluded endpoint. Prove a proposed supremum using both the upper-bound and approximation requirements rather than only drawing it.

Chapter 2: Sequences and Series

Convergence of a sequence is expressed through eventual closeness, with the index threshold allowed to depend on the desired accuracy. Limit laws require their hypotheses, and boundedness alone does not imply convergence. Monotone convergence, subsequences, limit superior and inferior, and Bolzano–Weierstrass organize different ways a sequence can behave. Cauchy sequences describe internal closeness and connect with completeness. A series concerns partial sums, not merely the limit of its terms. Absolute and other convergence questions require appropriate arguments and should not be settled by a few computed sums. The chapter's later series material extends the toolkit while preserving the same emphasis on precise conditions and estimates.

Study application: Prove convergence of a simple sequence directly from the definition, giving an explicit index bound. Then compare a divergent series whose terms approach zero with a convergent series, identifying the argument that distinguishes their partial sums.

Chapter 3: Continuous Functions

Limits of functions concern values near a point within the domain, while continuity connects that behavior with the value at the point. Sequence-based reasoning links this chapter with the previous one. Extreme-value and intermediate-value results depend on particular domain and continuity assumptions; their conclusions cannot be carried to arbitrary sets or discontinuous functions. Uniform continuity changes the dependence of the allowable input distance, making quantifier order a substantive issue. Limits at infinity and monotone functions extend the analysis beyond one local picture. A good study approach pairs every theorem with a counterexample showing why an important hypothesis cannot simply be removed.

Study application: Write continuity and uniform-continuity statements with their quantifiers. Apply an existence theorem on a suitable interval, then construct or examine a counterexample after removing continuity or the required domain condition.

Chapter 4: The Derivative

The derivative is defined through a limit of difference quotients and supports local linear reasoning. Differentiability implies continuity at the relevant point, but the converse fails. The Mean Value Theorem connects local derivatives with changes across an interval under specific endpoint and interior assumptions. Taylor's Theorem supplies approximation together with a remainder statement and appropriate regularity requirements. The one-variable inverse-function result explains when a local inverse inherits differentiability, rather than assuming every invertible continuous function has a well-behaved derivative. Throughout, routine differentiation should be distinguished from proving that a derivative exists or that a theorem applies. A computation is strongest when its domain and conditions are explicit.

Study application: Use the Mean Value Theorem to prove an inequality or monotonicity claim after checking all hypotheses. For a Taylor approximation, state the remainder information needed to justify an error estimate and distinguish it from a purely formal expansion.

Chapter 5: The Riemann Integral

The text develops the Riemann integral through upper and lower Darboux sums. Boundedness, partitions, suprema, and infima turn accumulation into a precise definition. Integrability requires the appropriate agreement of upper and lower constructions, not simply the existence of a plausible area sketch. Integral properties and the Fundamental Theorem of Calculus connect this construction with earlier continuity and derivative results under stated conditions. The chapter also develops logarithm and exponential functions and treats improper integrals through limits. An improper integral's convergence is a separate issue from ordinary integrability on each finite subinterval. Keeping the two versions of the fundamental theorem and their hypotheses distinct prevents unsupported differentiation or antidifferentiation arguments.

Study application: Estimate the gap between upper and lower sums for a simple function and use it in an integrability proof. Compare with a bounded function whose behavior on every subinterval prevents that gap from shrinking, then state why boundedness alone is insufficient.

Chapter 6: Sequences of Functions

A sequence of functions introduces more than one limiting variable. Pointwise convergence permits the convergence threshold to depend on the point, while uniform convergence controls the whole stated domain at once. This difference affects whether continuity and other operations can pass to a limit. Interchanging limits, integration, or differentiation requires appropriate additional hypotheses rather than a general rule that limits commute. Picard's theorem provides an application in which successive approximations, estimates, and completeness help establish existence and uniqueness for an ODE under suitable conditions. The chapter synthesizes earlier ideas and shows why careful control of an approximation can matter more than a persuasive graph.

Study application: Analyze the sequence x to the nth power on the unit interval and on a shorter closed interval ending below one. Identify the pointwise limit and compare uniform convergence, explaining how the domain changes the estimate.

Chapter 7: Metric Spaces

Metric spaces abstract distance while preserving ideas of convergence and continuity. Open and closed sets are relative to the ambient space, so a set's classification can change when the space changes. Cauchy sequences and completeness differ from compactness. Familiar finite-dimensional closed-and-bounded criteria should not be assumed valid in every metric space. Compactness and continuous maps support existence and uniformity results, while contraction-based fixed-point reasoning uses completeness and a suitable self-map. The chapter revisits Picard's theorem from a more abstract perspective, showing how the earlier real-line arguments fit a broader framework. Good abstraction retains examples and nonexamples so definitions do not become disconnected symbols.

Study application: Compare completeness and compactness in the real line and in an infinite discrete metric example. For a proposed contraction argument, identify the space, metric, self-map property, contraction bound, and completeness assumption separately.

Study instructions

1. Review the introductory set and proof material before beginning limit arguments. Write quantifiers explicitly until their dependence is clear.
2. For every theorem, record hypotheses, conclusion, proof strategy, and one example showing the importance of a hypothesis.
3. Distinguish existence proofs from calculations and illustrations. A graph or numerical experiment can guide a conjecture without proving it.
4. Build epsilon-style arguments by deciding what must be bounded and then choosing the required index or distance from that target.
5. Keep sequence limits, function limits, pointwise convergence, uniform convergence, and interchange results separate.
6. Use counterexamples constructively: change one assumption and identify exactly which conclusion fails.
7. Read the real-line chapters before treating metric spaces as merely new vocabulary. Map each abstract definition back to concrete examples.
8. Attempt exercises before consulting hints or other sources and write a complete proof after finding the idea, checking every use of a theorem.

Questions for reflection

• Why can a set have a supremum without a maximum?
• How does completeness enter a convergence argument?
• What changes when quantifiers are reordered in a continuity definition?
• Why do terms approaching zero not guarantee convergence of a series?
• Which hypotheses permit an operation to pass through a limit?
• Why are completeness and compactness different properties?

Sources and edition coverage

Coverage follows the complete 312-page author-hosted Volume I, sixth edition, version 6.3 dated May 15, 2026: the introduction and all seven chapters. Further Reading, the index, and notation list support the main text. The author name is corrected to Jiří Lebl, and the page title identifies Volume I while preserving the URL. The authentic version 6.3 PDF title page is used as the cover instead of implying an older catalog image is edition-specific. This is an original reading guide. Consult the source for its full explanations, cases, worked examples, exercises, illustrations, and references. Study applications above are new practice suggestions.

Read the source edition · Catalog record