Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Let \(S=\{x\in\mathbb{R}:x^2<2\}\). Using completeness, prove that \(\alpha=\sup S\) satisfies \(\alpha^2=2\).
Timer not started. Twenty minutes of real work before a solution.
\(S\) is nonempty and bounded above, so \(\alpha\) exists. If \(\alpha^2<2\), a small enough positive increment stays in \(S\), contradicting the least-upper-bound property. If \(\alpha^2>2\), a slightly smaller number is still an upper bound. The only remaining possibility is \(\alpha^2=2\). Write the increments explicitly: if \(\alpha^2<2\), choose \(h>0\) with \(h<1\) and \(h<(2-\alpha^2)/(2\alpha+1)\).
Attack 2
Prove that a convergent sequence is Cauchy.
Timer not started. Twenty minutes of real work before a solution.
If \(a_n\to L\), take \(\varepsilon/2\). Past \(N\), both \(a_n\) and \(a_m\) are within \(\varepsilon/2\) of \(L\), so they are within \(\varepsilon\) of each other by the triangle inequality.
Attack 3
Show that \(f(x)=1/x\) on \((0,1]\) is continuous but not uniformly continuous.
Timer not started. Twenty minutes of real work before a solution.
Continuity at each \(c\in(0,1]\) is the usual reciprocal estimate. Not uniform: take \(x_n=1/n\), \(y_n=1/(n+1)\). The gaps go to 0 while \(|f(x_n)-f(y_n)|=1\). So no single \(\delta\) serves \(\varepsilon=1/2\).
Real analysis is the repair job on calculus. Every limit, derivative, and integral you have used is given hypotheses and a proof, and the places the proof fails become the famous counterexamples: a continuous nowhere-differentiable function, a sequence of continuous functions whose limit is discontinuous, a derivative that is not integrable in the Riemann sense on a fat set. The spine is the completeness of the real line, sequential compactness, uniform continuity on compact intervals, the Riemann integral, and uniform convergence.
Rudin’s Principles of Mathematical Analysis is the ceiling. The third-edition ISBN catalogued for it is 978-0-07-054235-8. It is paid, and its exercises are qualifying-exam material. You should not open Rudin as the first book. Jiří Lebl’s Basic Analysis is free, careful, and enough to learn the subject. Abbott’s Understanding Analysis, ISBN 978-1-4939-2711-1, is the paid book people love because the motivation is written in English; use it if you buy one. Pugh, ISBN 978-3-319-17770-0, is the other paid route, with harder exercises. MIT 18.100 is the course with problem sets.
ε–δ is not a style. It is the definition. If you translate it into a picture and then stop writing inequalities, you have not done the proof.
Core results
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Completeness. Every nonempty subset of \(\mathbb{R}\) that is bounded above has a least upper bound. Equivalent forms: every Cauchy sequence converges; every monotone bounded sequence converges; nested closed intervals have nonempty intersection.
From the least-upper-bound property to monotone convergence: the supremum of the set of terms is the limit. Cauchy completeness from monotone convergence uses a subsequence of dyadic windows. You should write one direction in full and know the names of the others.
Continuity. Continuity at a point is the ε–δ definition. On a compact set, a continuous real function is uniformly continuous, bounded, and attains its max and min. The intermediate value theorem holds on an interval.
Uniform continuity on \([a,b]\): if it failed, sequences with gaps going to 0 and values separated by a fixed ε would have a convergent subsequence by Bolzano–Weierstrass, and continuity at the limit point would be contradicted. The extreme value theorem is the same compactness plus the fact that the image of an interval is an interval.
Differentiation. Differentiability implies continuity. The mean value theorem holds. There exist functions continuous everywhere and differentiable nowhere; you should know that Weierstrass built one, and be able to state what that forbids you from claiming.
Differentiability implies continuity because the difference \(f(x)-f(a)=f'(a)(x-a)+o(x-a)\) goes to 0. The mean value theorem is the one you already proved in calculus, now with hypotheses you do not skip. The nowhere-differentiable example is a uniform limit of a series of jagged continuous functions; read one construction in Lebl or Abbott and rewrite the estimate.
Riemann integral. A bounded function on \([a,b]\) is Riemann integrable if and only if for every \(\varepsilon>0\) there is a partition whose upper and lower sums differ by less than \(\varepsilon\). Continuous functions are integrable. A function with finitely many discontinuities is integrable. The fundamental theorem needs continuity of \(f\), or integrability of \(f\) plus continuity of \(F\) in the other direction, and you should state which form you mean.
Uniform continuity gives a partition where the oscillation on every subinterval is small, hence upper minus lower is small. Finite discontinuities are isolated in small intervals whose total length makes their contribution less than ε.
Uniform convergence. A uniform limit of continuous functions is continuous. The Weierstrass M-test gives uniform convergence of a series. Power series converge uniformly on compact subsets of the open interval of convergence, so they may be integrated and differentiated termwise there.
The ε/3 argument: choose one function in the sequence within ε/3 of the limit everywhere, then use its continuity. The M-test compares with a convergent numerical series. Termwise integration of power series follows from uniform convergence on a smaller closed interval.
The ladder
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Tier 0 / First contact
Watch or read once. Do not study from this.
Tao, books page for Analysis I Read once, do not study from this. It is a prospectus for a construction of the reals. It tells you the subject is about building the line, not about computing more integrals.
Tier 1 / Main text
The thing you work through.
Jiří Lebl, Basic Analysis, and MIT 18.100A Free, current, and written as a course. The HTML home is https://www.jirka.org/ra/ . MIT 18.100A supplies another problem-set culture: https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ . Abbott, ISBN 978-1-4939-2711-1, is the paid book to buy if Lebl’s tone is too dry.
Tier 2 / Hard problems
The problem book.
18.100B and Pugh’s exercises 18.100B is the harder MIT track. Pugh, Real Mathematical Analysis, ISBN 978-3-319-17770-0, is paid and is the exercise upgrade. Do the 18.100 sets either way.
Tier 3 / The ceiling
The hardest questions that exist at this level.
Rudin, Principles of Mathematical Analysis, ISBN 978-0-07-054235-8 Baby Rudin is paid and is the qualifying-exam ceiling. No free legal PDF was verified. The link is 18.100B, the public problem set at that altitude. Work Rudin’s Chapter 1–8 exercises if you own the book. Do not read the solutions collections first.
Hardest questions
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
MIT 18.100B The harder track. Use it when 18.100A feels like translation rather than struggle.
Rudin, Chapters 1–8, the exercises Rudin is paid and unlinked as a file. This link is Lebl’s site, where you should be before you spend money. Rudin’s exercises are hard because the hints are absent and the theorems are stated at full strength.
Pass: written proofs, from memory, of: Cauchy sequences converge in \(\mathbb{R}\) (one direction, named), uniform continuity of a continuous function on \([a,b]\), and the ε/3 proof that a uniform limit of continuous functions is continuous. Plus ten Lebl exercises you got wrong on the first attempt, rewritten. Fail if any proof swaps the order of quantifiers.
Diagrams
Uniform continuity is one δ for the whole interval. Pointwise continuity is a δ that depends on the point.
Escalating sets
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Set 1. The line
Prove the archimedean property from the least-upper-bound property.
Construct \(\sqrt{2}\) as a supremum.
Prove every monotone bounded sequence converges.
Show that a set of reals with an upper bound has a least upper bound, by naming the axiom you use. Do not prove the axiom.
Give a bounded set of rationals with no rational least upper bound.
Set 2. Sequences
Prove convergent implies Cauchy.
Prove a Cauchy sequence is bounded.
Bolzano–Weierstrass: every bounded sequence has a convergent subsequence. Write it.
Give a sequence with two distinct limit points.
Show \((-1)^n\) does not converge, from the definition.
Set 3. Continuity and derivatives
ε–δ proof that \(x^2\) is continuous at 3.
Prove \(1/x\) is not uniformly continuous on \((0,1]\).
Prove the intermediate value theorem.
Prove the mean value theorem from Rolle, with hypotheses.
Show differentiability implies continuity.
Set 4. Integral and series of functions
Prove a continuous function on \([a,b]\) is Riemann integrable.
Give a function that is discontinuous at one point and still integrable.
Prove the ε/3 argument for uniform limits.
State the M-test and use it on \(\sum x^n/n^2\) on \([-1,1]\).
Show that term-by-term integration of a power series is valid inside the radius.
Set 5. Ceiling
Ten hard Lebl exercises, rewritten after you were stuck.
One 18.100 problem set, chosen from the middle of the course.
Rudin Chapter 2 exercises if you own it; otherwise the compactness section of Lebl, all of it.
Blank-page proof of Bolzano–Weierstrass.
A counterexample to interchanging limit and integral, written with the functions explicit.
Return bank
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
\(\sqrt{2}\) as a supremum.
Convergent implies Cauchy.
\(1/x\) not uniformly continuous on \((0,1]\).
Bolzano–Weierstrass.
Intermediate value theorem.
Continuous on a compact interval implies integrable.
ε/3 for uniform limits.
M-test on \(\sum x^n/n^2\).
Differentiable implies continuous.
One counterexample you had to rebuild twice.
Oral pass
State completeness, prove that continuous functions on compact intervals are uniformly continuous, and explain uniform versus pointwise convergence with an example. No notes.