# Real analysis: make the quantifiers do the work — Study worksheet

[Study guides](https://unireadinglist.com/guides/real-analysis/)

[Course topic: Real Analysis](https://unireadinglist.com/subjects/mathematics/#course-real-analysis)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

A first proof-based route through sequences, function limits and uniform convergence on a real interval. Lebl means the sixth major edition, volume I; Abbott the 2015 second edition; Bartle–Sherbert the 2011 fourth edition. This is single-variable analysis preparation, with no assumption of measure theory. Later metric-space or multivariable material is outside this six-output route.

## Choose your starting point

- [ ] I need a free proof course
  Choose Lebl after a basic introduction to proofs; arrange feedback because the author intentionally supplies no exercise solutions manual.

- [ ] I need motivation before formal detail
  Choose Abbott when discussion examples help you understand what a definition is protecting against.

- [ ] I want a systematic traditional spine
  Choose Bartle–Sherbert for a conventional progression from preliminaries and real numbers to limits and continuity.

## Compare the three books

### Basic Analysis: Introduction to Real Analysis

Jiří Lebl · 6
Free first analysis course

Choose it when: You can write quantified statements and want a complete volume without a purchase.
Use another route when: You need a complete public answer key or want multivariable calculus before real-line foundations.
Selected reading: Volume I: Sequences and Series, Continuous Functions, then Sequences of Functions; use Real Numbers for completeness.
Limits and trade-offs: No solutions manual: checking each implication and obtaining feedback are part of the work. Volume II is a continuation.

#### Try before choosing

Using a sequence-limit definition, prove 1/(n+1)→0 with an explicit index choice.
Check your work: Give N depending on epsilon, then prove the inequality for every n≥N; a list of decimals is insufficient.

Your notes:

________________________________

[Official free reading](https://www.jirka.org/ra/)

[Book record and access options](https://unireadinglist.com/subjects/mathematics/basic-analysis-lebl/)

### Understanding Analysis

Stephen Abbott · 2 · 2015
Motivated proof development

Choose it when: You need examples that make approximation and finite-to-infinite pitfalls concrete.
Use another route when: You want only computational calculus or expect instructor solutions to be freely available.
Selected reading: Sequences and Series, Functional Limits and Continuity, and Sequences and Series of Functions.
Limits and trade-offs: Discussion and projects reward slower investigation; they are not a shortcut around writing the proofs.

#### Try before choosing

For x^n on \[0,1\], write the pointwise limit and locate the failure of a uniform claim.
Check your work: Treat x=1 separately and exhibit points near 1; do not infer uniformity from a graph at fixed resolution.

Your notes:

________________________________

ISBN: 9781493927111

[Book record and access options](https://unireadinglist.com/subjects/mathematics/understanding-analysis/)

### Introduction to Real Analysis

Robert G. Bartle; Donald R. Sherbert · 4 · 2011
Systematic first-course reference

Choose it when: You prefer definitions and a regular theorem/exercise sequence across the real line.
Use another route when: You are looking for a graduate measure-theory or general multivariable-analysis text.
Selected reading: Preliminaries, The Real Numbers, Sequences and Series, Limits, Continuous Functions; later sequences of functions as required.
Limits and trade-offs: The conventional structure is easy to follow by topic, but you must supply examples that distinguish similar hypotheses.

#### Try before choosing

For f\_n below, produce an upper bound independent of x and explain the role of a supremum.
Check your work: The same N must control every x in the domain. A separate N(x) proves a weaker statement.

Your notes:

________________________________

ISBN: 978-0-471-43331-6

[Book record and access options](https://unireadinglist.com/subjects/mathematics/introduction-to-real-analysis/)

## Check your preparation

### Can I negate a quantified limit statement?

Your notes:

________________________________

### Can I manipulate inequalities before choosing N?

Your notes:

________________________________

### Can I use calculus without mistaking it for a proof of convergence?

Your notes:

________________________________

## A six-step study plan

- [ ] Write two definitions

Selected reading: Your chosen sequence/function-limit units.

What to do: Produce pointwise and uniform convergence definitions side by side, marking permitted dependencies.

Check your work: Only the pointwise definition permits N to depend on x.

Your notes:

________________________________

- [ ] Prove a scalar limit

Selected reading: Lebl Sequences and Series; Abbott or Bartle matching unit.

What to do: Write a complete epsilon proof for 1/(n+1).

Check your work: The final implication applies to every n≥N, not only n=N.

Your notes:

________________________________

- [ ] Find the global error

Selected reading: Your chosen derivative and sequences-of-functions units.

What to do: Differentiate x^n(1−x), check endpoints and produce its exact maximum on \[0,1\].

Check your work: An interior critical point alone does not establish the maximum.

Your notes:

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- [ ] Convert a bound into a tolerance

Selected reading: Sequences of functions.

What to do: Use a bound independent of x to provide a valid common N for the worked example.

Check your work: The proof explicitly includes x=0 and x=1.

Your notes:

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- [ ] Stress-test the hypothesis

Selected reading: The same uniform-convergence unit.

What to do: Solve the n-scaled follow-up; give both a fixed-point argument and a moving-point obstruction.

Check your work: A nonzero lower bound along x\_n defeats uniformity even though each fixed point converges.

Your notes:

________________________________

- [ ] Review a false proof

Selected reading: Definitions and your own two solutions.

What to do: Annotate the claim “each x has a suitable N, so take one N for all x.”

Check your work: Identify the unjustified quantifier interchange and replace it by a supremum estimate or counterexample.

Your notes:

________________________________

## Worked exercise: A moving maximum can still shrink uniformly

For integers n≥1, let f\_n(x)=x^n(1−x) on \[0,1\]. Find the pointwise limit and the exact maximum error. Prove uniform convergence to zero and give a common index for a tolerance epsilon\>0.

Assumptions: Real-valued functions on the whole closed interval \[0,1\]; ordinary pointwise and supremum-norm convergence. Each n is a positive integer.

Your notes:

________________________________

## Change an assumption: A growing prefactor changes the quantifier problem

Keep the domain \[0,1\] but replace f\_n by g\_n(x)=n x^n(1−x). Determine pointwise and uniform convergence to zero. Test x\_n=1−1/(2n), and justify the fixed-point limit rather than assuming it.

Your notes:

________________________________

## Before moving on

- Write a quantified epsilon proof with a common index.
- Explain why the scaled family defeats uniformity without defeating pointwise convergence.
- Choose a text and a proof-feedback plan that match your readiness.

## Answer and interpretation

### Can I negate a quantified limit statement?

Write the order of epsilon, N, n and x. Uniform failure means one positive tolerance defeats every proposed common N.

### Can I manipulate inequalities before choosing N?

For 1/(n+1)\<epsilon, choose N=ceil(1/epsilon). For n≥N the inequality is strict; keep epsilon positive.

### Can I use calculus without mistaking it for a proof of convergence?

A derivative can locate a global maximum after endpoints and derivative signs are checked. Uniform convergence still needs a bound tending to zero.

### A moving maximum can still shrink uniformly

1. For fixed 0≤x\<1, x^n→0 and 0≤1−x≤1; at x=1 the function is identically zero. The pointwise limit is therefore zero everywhere.
2. For 0\<x\<1, f\_n′(x)=x^(n−1)\[n−(n+1)x\]. It is positive before x=n/(n+1) and negative after it; both endpoints give zero. The global maximum is n^n/(n+1)^(n+1).
3. Rewrite that maximum as (n/(n+1))^n/(n+1)≤1/(n+1). Choose N=ceil(1/epsilon); for every n≥N and every x, |f\_n(x)|≤1/(n+1)\<epsilon.

The pointwise and uniform limits are both zero. The maximum is attained at x=n/(n+1), with value n^n/(n+1)^(n+1). N=ceil(1/epsilon) is a valid common index.

Test the result: For n=1 the maximum is 1/4 at x=1/2. For n=2 it is 4/27 at x=2/3. The maximizing point approaches 1, but its height tends to zero; motion of a peak alone does not disprove uniform convergence.

### Change an assumption: A growing prefactor changes the quantifier problem

1. At x=0 and x=1 the value is zero. For fixed 0\<x\<1, set a\_n=n x^n. Its successive ratio is x(1+1/n). Choose q with x\<q\<1; eventually the ratio is at most q, so a\_n is bounded by a decaying geometric sequence. Thus g\_n(x)→0 at each fixed point.
2. At x\_n=1−1/(2n), g\_n(x\_n)=(1/2)(1−1/(2n))^n. Bernoulli’s inequality gives (1−1/(2n))^n≥1−n/(2n)=1/2, so this value is at least 1/4 for every n.
3. Take epsilon=1/8. For every proposed N, choose n≥max(N,1) and the corresponding x\_n. Its error exceeds epsilon, so no common index controls the entire interval.

g\_n converges pointwise to zero but not uniformly on \[0,1\]. The moving points retain an error of at least 1/4.

Test the result: For n=1, x\_1=1/2 and g\_1(x\_1)=1/4. The obstruction is a sequence of changing points; it does not contradict convergence at any fixed x. A finite numerical grid may miss a narrow peak.

## Common mistakes

### An index with hidden dependence

Writing N(x,epsilon) and later calling it a common N changes the claim. Keep dependence visible.

### Only checking stationary points

Compare endpoints and derivative signs before reporting a global maximum.

### Using a finite plot as a proof

A fixed grid cannot rule out peaks moving between samples as n changes. Use a bound or an explicit witness.

## Sources and edition checks

- [Author: Basic Analysis, version and contents](https://www.jirka.org/ra/)
  The author identifies the two-volume scope, basic-proof prerequisite, sixth-edition revision and deliberate absence of an exercise solutions manual.

- [Springer: Understanding Analysis, second edition](https://link.springer.com/book/10.1007/978-1-4939-2712-8)
  Springer distinguishes the 2015 hardcover second edition and provides the named sequence, limit and function-series contents.

- [Wiley: selected fourth-edition hardcover](https://www.wiley.com/en-nl/introduction-to-real-analysis-4th-edition-p-9780471433316)
  Wiley identifies the exact 2011 fourth-edition hardcover and the systematic real-line progression; it is an introduction to rigorous analysis.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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