Basic Analysis: Introduction to Real Analysis PDF Download – Jiri Lebl (BS Math)

BS Mathematics students taking Real Analysis (sometimes titled Introduction to Analysis or Modern Analysis) can download Basic Analysis: Introduction to Real Analysis, Volume I by Jiri Lebl, free from the author’s own official site under a dual Creative Commons licence. It is a proof-based, one-semester textbook that builds analysis from the ground up — starting from the properties of the real numbers and working through sequences, continuity, differentiation, the Riemann integral, sequences of functions, and metric spaces.

This book is aimed at BS Math students preparing for their Real Analysis board and semester exams, where every claim needs a proof and a hand-wave is not enough. Each of its eight chapters ends with a real exercise set — 539 problems in total, ranging from routine epsilon–delta verifications to full theoretical proofs — making the book equally useful for first-pass learning of rigorous analysis and for exam revision closer to test day.

Book Overview

Chapter List

Chapter 0: Introduction — Sets, Induction, and Cardinality

Difficulty: Medium · Key topics: mathematical induction, set operations, functions, equivalence relations, countable and uncountable sets, Cantor’s theorem

This opening chapter is a fast review of the toolkit every later chapter assumes, not new analysis. It restates the Principle of Mathematical Induction with a full proof, then works through set operations, functions (injective, surjective, bijective), and equivalence relations and the partitions they induce. It closes with cardinality: what it means for two sets to have the same size via a bijection, why the rational numbers are countable while the real numbers are not (Cantor’s diagonal argument), and why no set has a bijection with its own power set (Cantor’s theorem).

Key Points:

• Mathematical induction has two parts: the basis statement P(1), and the induction step — P(n) true implies P(n+1) true
• A set is countable if there is a bijection between it and a subset of the natural numbers (finite sets and countably infinite sets are both called countable)
• The rational numbers are countable; the real numbers are uncountable, proved by Cantor’s diagonal argument
• Cantor’s theorem: for any set S, there is no bijection between S and its power set P(S)
• A function is invertible if and only if it is both one-to-one (injective) and onto (surjective)

Memory Tip: Treat this chapter as a checklist, not new material — if a term like bijection or equivalence class feels unfamiliar, stop and master it here, because every later chapter uses this vocabulary without re-explaining it.

Common Mistake: Skipping straight to Chapter 1 without working through the induction and cardinality proofs here, then getting stuck when Chapter 1 uses induction to prove basic facts about the real numbers and Chapter 2’s Bolzano–Weierstrass argument assumes comfort with countability.

Important Questions:

• What are the two parts of a proof by mathematical induction? The basis statement (P(1) is true) and the induction step (if P(n) is true, then P(n+1) is true).
• Is the set of rational numbers countable, and is the set of real numbers? The rationals are countable; the reals are uncountable, which the chapter proves using Cantor’s diagonal argument.

Chapter 1: The Real Numbers

Difficulty: Medium · Key topics: ordered fields, least upper bound property, supremum and infimum, absolute value, triangle inequality, Archimedean property, decimal representation

Rather than constructing the real numbers from scratch, this chapter lists the properties that characterize them as a complete ordered field and works from there. The extra property beyond the rational numbers is the least upper bound property: every nonempty subset of the reals that is bounded above has a supremum in the reals — and the rationals do not have this property, which is exactly why they are not enough for analysis. The chapter proves the Archimedean property, defines absolute value formally and proves the triangle inequality, studies intervals and shows the real numbers are uncountable, and closes by proving every real number has a decimal representation.

Key Points:

• The real numbers form a complete ordered field: every nonempty subset bounded above has a supremum (least upper bound) in the real numbers
• Archimedean property: for any real x > 0 and any real y, there is a natural number n with nx > y
• Triangle inequality: |a + b| ≤ |a| + |b| for all real a, b
• The set {x in Q : x² < 2} has rational upper bounds but no least one in Q — exactly where the rationals fail to have the least upper bound property
• A decimal representation of a real number is non-unique only for numbers of the form m / 10^n

Memory Tip: “Suprema and infima are guaranteed to exist in the reals, not in the rationals” is the one sentence that separates analysis from ordinary algebra — nearly every later existence theorem traces back to this completeness property.

Common Mistake: Confusing a supremum with a maximum — a bounded set need not contain its supremum (the interval (0,1) has supremum 1, but 1 is not in the set), while a maximum, when one exists, must be an element of the set itself.

Important Questions:

• What is the least upper bound property, and why does the set of rational numbers not have it? Every nonempty subset of the reals bounded above has a supremum in the reals; the rationals lack this because a set like {x in Q : x² < 2} has rational upper bounds but no least one in Q.
• State the Archimedean property. For any real x > 0 and any real y, there exists a natural number n such that nx > y.

Chapter 2: Sequences and Series

Difficulty: Hard · Key topics: convergence, monotone convergence theorem, limit superior and inferior, Bolzano–Weierstrass theorem, Cauchy sequences, series convergence, rearrangement

This is the book’s longest chapter (100 exercises) and the one that fully activates the epsilon–N definition of a limit. It proves the standard limit laws for sums, products, and quotients of convergent sequences, the monotone convergence theorem (a monotone sequence converges if and only if it is bounded), and limsup/liminf as the largest and smallest limits of any subsequence. The Bolzano–Weierstrass theorem — every bounded sequence has a convergent subsequence — is proved using limsup. Cauchy sequences are introduced as a way to prove convergence without knowing the limit in advance, and the chapter closes with series: convergence, absolute versus conditional convergence, and the surprising fact that a conditionally convergent series can be rearranged to converge to any real number at all.

Key Points:

• Monotone convergence theorem: a monotone sequence is bounded if and only if it converges
• limsup of a sequence is the largest limit of any of its convergent subsequences; liminf is the smallest
• Bolzano–Weierstrass theorem: every bounded sequence of real numbers has a convergent subsequence
• A sequence of real numbers converges if and only if it is a Cauchy sequence (completeness of the real numbers)
• A conditionally convergent series can be rearranged to converge to any real number, or to diverge — only absolute convergence guarantees the sum is rearrangement-invariant

Memory Tip: Cauchy sequences let you prove convergence “from inside” the sequence, without ever naming the limit — reach for the Cauchy criterion whenever a recursively defined sequence needs a convergence proof.

Common Mistake: Assuming limsup and liminf always equal the sequence’s limit — that equality holds only once the sequence actually converges; for a general bounded sequence they can differ, and that gap is exactly what the Bolzano–Weierstrass proof exploits.

Important Questions:

• What does the Bolzano–Weierstrass theorem state? Every bounded sequence of real numbers has a convergent subsequence.
• Why is showing a sequence is Cauchy useful for proving convergence? Because in the real numbers a sequence converges if and only if it is Cauchy, so the Cauchy criterion proves convergence without needing to know the limit value in advance.

Chapter 3: Continuous Functions

Difficulty: Medium · Key topics: limits of functions, continuity, extreme value theorem, intermediate value theorem, uniform continuity, monotone functions

Continuity is defined first through the limit of a function at a point, then shown equivalent to a sequential characterization: f is continuous at c exactly when f(x_n) converges to f(c) for every sequence x_n converging to c. Two headline results follow for continuous functions on a closed, bounded interval: the Extreme Value Theorem (a continuous function attains a maximum and a minimum) and the Intermediate Value Theorem, credited to Bolzano, guaranteeing f takes every value between f(a) and f(b). The chapter then separates plain continuity from the stronger uniform continuity, proves that continuity on a closed bounded interval automatically upgrades to uniform continuity, and closes with monotone functions, whose only possible discontinuities are jumps.

Key Points:

• f is continuous at c if and only if f(x_n) converges to f(c) for every sequence x_n converging to c in the domain (sequential continuity)
• Extreme Value Theorem: a continuous function on a closed, bounded interval attains both a maximum and a minimum value
• Intermediate Value Theorem (Bolzano): if f is continuous on [a,b] and y lies strictly between f(a) and f(b), then f(c) = y for some c between a and b
• A continuous function on a closed, bounded interval is automatically uniformly continuous
• A monotone function can only have jump discontinuities, never any other kind

Memory Tip: When a proof needs “inputs close enough anywhere on the interval give outputs close enough, with one delta that works everywhere,” that is uniform continuity, not plain continuity — closed and bounded is exactly the condition that upgrades one to the other.

Common Mistake: Applying the Intermediate Value Theorem without first checking the function is continuous on the whole closed interval — a single point of discontinuity anywhere in [a,b] invalidates the guarantee.

Important Questions:

• What does the Intermediate Value Theorem guarantee? If f is continuous on [a,b] and y is strictly between f(a) and f(b), there is some c between a and b with f(c) = y.
• Why is a continuous function on a closed, bounded interval automatically uniformly continuous? Because a closed, bounded interval is compact, and the chapter proves that continuity on a compact set always strengthens to uniform continuity.

Chapter 4: The Derivative

Difficulty: Medium · Key topics: differentiability, Rolle’s theorem, mean value theorem, Cauchy’s mean value theorem, Taylor’s theorem, Darboux’s theorem, inverse function theorem

The derivative is defined as a limit of a difference quotient, and this chapter proves the differentiation rules — sum, product, quotient, chain rule — rigorously rather than assuming them from calculus. Rolle’s theorem (a differentiable function with equal endpoint values has a point with derivative zero in between) is proved first and used to derive the Mean Value Theorem and its generalization, Cauchy’s Mean Value Theorem. Two shorter, optional sections follow: Taylor’s theorem with the Lagrange remainder, and Darboux’s theorem, the surprising fact that a derivative — even a discontinuous one — always satisfies the intermediate value property.

Key Points:

• The derivative f′(c) is defined as the limit, as x approaches c, of the difference quotient (f(x) − f(c)) / (x − c)
• Rolle’s theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then f′(c) = 0 for some c between a and b
• Mean Value Theorem: there exists c between a and b with f(b) − f(a) = f′(c)(b − a)
• Cauchy’s Mean Value Theorem generalizes the MVT to a ratio of two functions and underlies L’Hopital’s rule
• Darboux’s theorem: a derivative f′ always has the intermediate value property, even when f′ itself is not continuous

Memory Tip: Rolle’s theorem is the special case of the Mean Value Theorem where the two endpoint values happen to be equal — prove Rolle first, then tilt the picture with a linear correction term to get the full Mean Value Theorem.

Common Mistake: Assuming a function’s derivative must itself be continuous — Darboux’s theorem shows a derivative always satisfies the intermediate value property regardless, so a discontinuous derivative is unusual but not impossible.

Important Questions:

• State the Mean Value Theorem. If f is continuous on [a,b] and differentiable on (a,b), there exists c between a and b such that f(b) − f(a) = f′(c)(b − a).
• What does Darboux’s theorem say about derivatives? Even when f′ is not continuous, it still satisfies the intermediate value property — it takes every value between f′(a) and f′(b).

Chapter 5: The Riemann Integral

Difficulty: Hard · Key topics: Darboux sums, Riemann integrability, properties of the integral, Fundamental Theorem of Calculus, logarithm and exponential, improper integrals

This book defines the Riemann integral using upper and lower Darboux sums rather than tagged Riemann sums — a choice the author’s own introduction calls far more appropriate for a course at this level. A function is Riemann integrable exactly when its upper and lower integrals agree, and the chapter proves that both continuous functions and monotone functions are always integrable. Both forms of the Fundamental Theorem of Calculus are proved: one relates the integral to any antiderivative, and the other shows the integral function F(x), defined as the integral of f from a to x, is itself differentiable at every point where f is continuous. Optional sections rigorously define the exponential and logarithm from the integral and study improper integrals over unbounded intervals or unbounded functions.

Key Points:

• A bounded function is Riemann integrable on [a,b] exactly when its upper Darboux sum and lower Darboux sum can be made arbitrarily close — the upper and lower integrals agree
• Every continuous function on [a,b] is Riemann integrable, and so is every monotone function
• Fundamental Theorem of Calculus (first form): if F is an antiderivative of the integrable function f, the integral of f from a to b equals F(b) − F(a)
• Fundamental Theorem of Calculus (second form): F(x), the integral of f from a to x, is continuous everywhere and differentiable with F′(x) = f(x) wherever f is continuous
• The Darboux approach never picks sample points inside each subinterval, unlike the tagged-partition Riemann sum definition usually seen first in calculus

Memory Tip: Picture “the upper sum squeezes down, the lower sum squeezes up” — integrability is exactly the statement that these two squeezed values meet in the middle.

Common Mistake: Treating “integral” and “antiderivative” as the same thing — the integral is defined independently, as a limit of Darboux sums (informally, the area under the curve), and the Fundamental Theorem of Calculus is the nontrivial result connecting the two, not a definition.

Important Questions:

• When is a bounded function Riemann integrable on [a,b]? Exactly when its upper Darboux sum and lower Darboux sum can be made arbitrarily close, i.e. the upper and lower integrals are equal.
• What does the second form of the Fundamental Theorem of Calculus say? For an integrable function f, F(x) (the integral of f from a to x) is continuous on [a,b], and at any point where f is continuous, F is differentiable there with F′(x) = f(x).

Chapter 6: Sequences of Functions

Difficulty: Hard · Key topics: pointwise vs. uniform convergence, interchange of limits, term-by-term differentiation, Picard’s theorem

This chapter tackles what the book’s own introduction calls a central theme of modern analysis: when can two limiting operations be interchanged? It distinguishes pointwise convergence of a sequence of functions from the strictly stronger uniform convergence, and shows uniform convergence is exactly what is needed to swap a limit with an integral, or, under an extra hypothesis on the derivatives, with a derivative. The chapter closes with Picard’s theorem on the existence and uniqueness of solutions to ordinary differential equations, built from nearly every earlier tool in the book and singled out in the author’s own introduction as the signature result a first analysis course should build toward.

Key Points:

• A sequence of functions f_n converges uniformly to f on a set if the worst-case gap between f_n and f over the whole set goes to zero, not just the gap at each individual point
• Pointwise convergence does not preserve continuity in general; uniform convergence of continuous functions does, and the limit is continuous
• If f_n converges uniformly to f and each f_n is Riemann integrable, the limit of the integrals of f_n equals the integral of f
• Term-by-term differentiation of a function sequence needs the sequence of derivatives to converge uniformly, not just the original functions
• Picard’s theorem guarantees a unique local solution to an ordinary differential equation with an initial condition, proved using a sequence of successive approximations

Memory Tip: When a proof needs to swap a limit with an integral or a derivative, uniform convergence is the passport that allows it — pointwise convergence alone is not enough, and counterexamples are easy to build.

Common Mistake: Assuming that if f_n converges pointwise to f and every f_n is continuous, then f must be continuous too — this is false in general; only uniform convergence guarantees the limit inherits continuity.

Important Questions:

• What is the difference between pointwise and uniform convergence of a sequence of functions? Pointwise means the gap between f_n(x) and f(x) shrinks to zero at each fixed x separately; uniform means the worst-case gap over the whole domain shrinks to zero.
• Why does uniform convergence matter for integrals? If f_n converges uniformly to f and each f_n is Riemann integrable, the limit of the integrals of f_n equals the integral of f — an interchange that can fail under pointwise convergence alone.

Chapter 7: Metric Spaces

Difficulty: Hard · Key topics: metric spaces, open and closed sets, completeness, compactness, continuity on metric spaces, contraction mapping (fixed point) theorem

The final chapter generalizes everything the book has built for the real numbers to an abstract set equipped with a distance function (a metric), so the same limit arguments apply to spaces of functions, not just numbers. It defines open and closed sets, proves that compactness (every open cover has a finite subcover) is equivalent to sequential compactness (every sequence has a convergent subsequence) in a metric space, and reproves continuity in purely topological terms: f is continuous exactly when the preimage of every open set is open. The chapter ends by proving the Contraction Mapping (Banach Fixed Point) Theorem and using it to reprove Picard’s theorem from Chapter 6 by a completely different, more general method.

Key Points:

• A metric space is a set together with a distance function satisfying non-negativity, symmetry, and the triangle inequality
• In a metric space, a set K is compact if and only if every sequence in K has a subsequence converging to a point in K (compactness equals sequential compactness)
• A function between metric spaces is continuous if and only if the preimage of every open set is open — no epsilon–delta needed
• If a metric space is compact and a function on it is continuous, that function is automatically uniformly continuous — the same theorem from Chapter 3, now proved in full generality
• Contraction Mapping Theorem: a contraction on a complete metric space has a unique fixed point, found as the limit of repeated iteration from any starting point

Memory Tip: Translate every real-number-based definition as “replace the distance |x − y| with the metric d(x,y)” on a first pass — nearly every proof from Chapters 2 and 3 carries over to a general metric space almost word for word once that substitution is made.

Common Mistake: Assuming “closed and bounded” always implies compact, the way the Heine–Borel theorem states it does in the real numbers — the book itself proves this equivalence is special to R^n; in a general metric space, compactness must be checked directly, not inferred from boundedness alone.

Important Questions:

• What does it mean for a subset K of a metric space to be compact? Every sequence in K has a subsequence that converges to a point in K (sequential compactness, which the chapter proves is equivalent to the open-cover definition).
• State the Contraction Mapping (Banach Fixed Point) Theorem. A contraction mapping on a complete metric space has exactly one fixed point, which can be found as the limit of iterating the map from any starting point.

Download Basic Analysis: Introduction to Real Analysis PDF (Free)

This book is free from its official source, author Jiri Lebl’s own site. Click below to go to the official page, where you can get the Volume I PDF, browse the HTML edition, or grab the LaTeX source directly — no cost, no registration. The same page also links to Volume II for students continuing into a year-long, two-semester sequence.

How to Study This Book

Read Chapter 0 as review, not as new material. If your proof-writing course already covered sets, induction, functions and equivalence relations properly, skim it; if it did not, slow down here — the induction proofs in Chapter 1 and the cardinality argument behind Bolzano–Weierstrass in Chapter 2 both lean on this chapter directly.

For a standard one-semester Real Analysis course, the author’s own introduction suggests this path, and it is worth following closely: all of Chapter 1 (Real Numbers), Sections 2.1 through 2.5 of Chapter 2 (Sequences and Series — Section 2.6, More on Series, can be safely skipped or covered only partially if time is short), Sections 3.1 through 3.4 of Chapter 3 (Continuous Functions — Uniform Continuity and Monotone Functions are optional add-ons), Sections 4.1 and 4.2 of Chapter 4 (The Derivative — Taylor’s Theorem and the Inverse Function Theorem are both explicitly optional sections), and Sections 5.1 through 5.3 of Chapter 5 (The Riemann Integral — the Logarithm/Exponential and Improper Integrals sections are optional and depend on the optional sections from earlier chapters).

If your course also covers metric spaces (common in a more rigorous or faster-paced offering), the book’s own suggested path compresses the earlier chapters slightly and adds Sections 7.1 through 7.6 of Chapter 7 in their place, largely skipping Chapter 6.

Picard’s theorem — proved once without metric spaces at the end of Chapter 6, and proved again more generally using the Contraction Mapping Theorem at the end of Chapter 7 — is explicitly marked as an optional, can-be-skipped topic in both places, but it is the closest thing this book has to a final exam-worthy showpiece: it uses sequences, uniform convergence, completeness, and continuity together in one proof. If your syllabus has room for it, it is worth the extra lecture.

What this book does not cover at all: measure theory, Lebesgue integration, and multivariable (several-variables) analysis. That material belongs entirely to Volume II, the sequel to this book by the same author, used for year-long two-semester Real Analysis sequences. If your course is a single semester and stops at metric spaces, Volume I on its own is complete for it.

A prior course in mathematical proof-writing is assumed throughout — Chapter 0’s own Section 0.3 is a fast review, not a first introduction to sets, logic, or induction.

Used In These Programs

This book is used for the Real Analysis course in: BS Mathematics. Browse all Real Analysis books or all Mathematics books.

Who Should Read This

This book is written for BS Mathematics students taking their first Real Analysis course, usually in the junior year, who have already completed a course in mathematical proof and rigor — a discrete mathematics course such as freebooks.pk’s own Discrete Mathematics: An Open Introduction covers exactly that groundwork (sets, logic, induction, basic proof technique) that Chapter 0 here only reviews quickly rather than teaches from scratch. It is not a good fit as a first exposure to proofs, and it is not a calculus textbook either — it assumes calculus computation is already familiar and instead rebuilds the same ideas (limits, continuity, derivatives, integrals) on a fully rigorous foundation. Students who found the Riemann integral or convergence of sequences confusing at the computational level in calculus will find this book’s careful, proof-first treatment of exactly those ideas clarifying rather than repetitive.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Mathematics, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

FAQs

Related Books

• Discrete Mathematics: An Open Introduction – Oscar Levin (4th Edition)
• Calculus Volume 1 – Strang & Herman (OpenStax)
• Calculus Volume 2 – Strang & Herman (OpenStax)

Basic Analysis: Introduction to Real Analysis, Volume I is this project’s first Real Analysis book — the core proof-based course most BS Mathematics students take in their junior year, rebuilding calculus on a fully rigorous foundation through sequences, continuity, derivatives, integrals, and metric spaces. Browse more Mathematics books for the rest of your degree.

Basic Analysis: Introduction to Real Analysis, Volume I, Version 6.3, by Jiri Lebl. Free, dual licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0) or the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). Access for free at https://www.jirka.org/ra/