Overview
Many strong students are thrown by their first proof-based course. Lectures present polished proofs that hide how they were found, and problem sets demand arguments that are both correct and properly written. We separate the two stages: discovery, where you experiment with examples, draw pictures and work backwards from the goal; and presentation, where you write a clear chain of justified statements.
We cover the techniques that appear across courses: direct proof, contrapositive, contradiction, weak and strong induction, case analysis, constructive and non-constructive existence proofs, counterexamples, and epsilon-delta and epsilon-N arguments in analysis. For each we show the logical structure, typical phrasing and the most common mistakes markers see.
Problem sets are for your learning and usually count toward your grade, so we do not solve them for you. Instead we work parallel problems, give hints that point to the key idea without completing the argument, and review your written proofs for logical gaps and presentation.
Who it’s for and the problems it solves
This is for you if you are…
- First- and second-year mathematics, computer science and physics students in their first proof-based courses
- Students in real analysis, linear algebra, abstract algebra, number theory, topology or discrete mathematics
- Master's students filling gaps before a qualifying exam or a theory-heavy program
- Economics and engineering graduate students facing mathematical proofs in theory courses
- Self-learners working through proof-based textbooks
Common problems we help with
- You read a proof and follow each line but could never have produced it yourself
- Your proofs come back marked 'not rigorous' or 'circular' and you do not see why
- Epsilon-delta definitions of limits and continuity feel impossible to use
- Induction proofs work for easy sums but collapse on inequalities or recursive definitions
- You do not know when you are allowed to say 'clearly' or cite a theorem
What’s included — and what isn’t
Included
- Logic foundations: quantifiers and their negation, implication, contrapositive and converse, set notation
- Proof techniques: direct, contrapositive, contradiction, cases, induction (weak, strong, structural), pigeonhole, counterexample
- Real analysis: epsilon-delta limits and continuity, sequences and series convergence, completeness, uniform continuity and convergence
- Linear algebra: vector spaces, linear independence, bases and dimension, linear maps, eigenvalues and proofs about them
- Abstract algebra: groups, subgroups, homomorphisms, Lagrange's theorem, rings and fields
- Discrete mathematics and number theory: combinatorial proofs, recurrences, divisibility, modular arithmetic, graph theory
- Proof reading and feedback: identifying gaps, hidden assumptions, circular reasoning and notation problems in your drafts
- Mathematical writing and LaTeX habits: defining notation, structuring lemmas, and typesetting with amsthm environments
- Problem-solving strategies from Pólya and competition practice: specializing, generalizing, invariants, extremal principle
Not included
- Writing solutions or proofs to graded problem sets, take-home exams or qualifying exams
- Taking online quizzes or timed assessments for anyone
- Producing proofs for thesis results as if they were the student's own work
What you receive
- Technique guides
- Short written guides for each proof technique covered, with the logical template, a worked example and a list of common errors.
- Parallel problem sets
- Practice problems similar to your course level but not taken from your assignments, with full solutions released after you attempt them.
- Proof feedback
- Line-by-line comments on proofs you have written, marking gaps, unjustified steps and clarity issues without supplying the missing argument.
- Hint ladders
- For your own study problems, a graduated series of hints that you reveal one at a time, so you get as much help as you need and no more.
How it works
Assess where you are
We ask you to prove a couple of short statements aloud. Your approach shows whether the gap is logic, definitions, technique or writing.
Teach the structure
We explain the technique's logical form, work an example together and discuss what makes it valid.
Guided discovery
On a parallel problem you do the thinking while we ask questions: what does the definition say, what would a counterexample look like, what do you need to show?
Write and review
You write the proof; we review it for correctness and clarity and you revise.
Build independence
Hints are reduced over time until you can start unfamiliar problems yourself.
Proof techniques we teach
- Mathematical induction
- Base case, inductive hypothesis and inductive step, with strong induction for recurrences and structural induction for recursively defined objects.
- Contradiction
- Assuming the negation and deriving an impossibility, as in the irrationality of √2 or the infinitude of primes, and how it differs from contrapositive.
- Epsilon-delta and epsilon-N
- Scratch work to find δ in terms of ε, then a clean forward proof; handling products, quotients and bounding tricks.
- Counterexamples
- Disproving universal statements with a single well-chosen example and explaining exactly which hypothesis it violates.
- Combinatorial and bijective proofs
- Counting the same set two ways or constructing a bijection, common in discrete mathematics courses.
- Quantifier discipline
- Writing out 'for all' and 'there exists' in order before starting, and negating them correctly.
Scratch work versus the final proof
An epsilon-delta proof is usually found backwards and written forwards. In scratch work you start from |f(x) − L| < ε and ask what δ would make it true; in the proof you start by choosing that δ and show the inequality follows. Mixing the two produces circular arguments, one of the most common reasons proofs lose marks. We make the separation explicit on paper and in LaTeX.
Mistakes markers see most often
Most lost marks come from a small number of recurring errors. Learning to spot them in your own drafts is quicker than memorizing more proofs.
- Proving the converse instead of the statement
- Checking a few examples and treating them as a proof
- Using the inductive hypothesis for n + 1 instead of n
- Choosing δ that depends on x when uniform continuity is required
- Dividing by a quantity that might be zero
- Introducing a symbol without saying what it is or where it lives
Using AI tools and online solutions carefully
Chat assistants and solution sites can produce proofs that look fluent but contain gaps or quietly assume what is to be proved, and using them on graded work may breach your course rules. We teach you to check any argument line by line against the definitions, which is also the skill examiners test.
What the work can look like
Illustrative example — fictional data
Sample proof feedback on an induction argument
A student practicing induction wrote a proof that 2ⁿ > n² for all n ≥ 5, using a textbook exercise that was not part of a graded assignment. The draft and comments are invented for illustration.
Base case
Correct: 32 > 25 for n = 5. Good that you checked the stated starting value rather than n = 1, where the claim is false.
Inductive step
You wrote 2ⁿ⁺¹ = 2·2ⁿ > 2n² and then 'so 2ⁿ⁺¹ > (n + 1)²'. The missing step is showing 2n² ≥ (n + 1)², which is equivalent to n² − 2n − 1 ≥ 0. State why this holds for n ≥ 5, for example by noting the larger root is 1 + √2 < 3.
Presentation
Write the inductive hypothesis explicitly, 'Assume 2ᵏ > k² for some k ≥ 5', and use k rather than n inside the step so the reader can see what is fixed.
Illustrative example with an invented student draft. We give this kind of feedback on practice proofs and on drafts where your course permits outside feedback.
What you’ll need to provide
- Course name, syllabus and textbook (for example Abbott, Rudin, Axler, Dummit and Foote, Velleman)
- Examples of proofs you have written and any marker feedback
- The definitions and theorems your course has covered so far, since what you may cite depends on them
- Your course's policy on collaboration and outside help with problem sets
You don’t need everything ready to ask for a quote — tell us what you have and we’ll say what’s missing.
What affects price and turnaround
Price depends on
- Course level, from introductory proofs to graduate analysis or algebra
- Number and length of sessions
- Volume of written proof feedback between sessions
- Preparation of custom parallel problem sets
Turnaround depends on
- Your exam dates and the pace of your course
- How many drafts you want reviewed each week
- Access to your course notes, so our notation matches
Every project is quoted individually. How we price our work.
Questions people ask before starting
Will you solve my problem set?
No. Problem sets are part of your assessment and your learning. We teach the techniques, work parallel problems, give graduated hints on study problems and review proofs you have written. If your course forbids any outside help on a problem set, we only work on other material.
I understand the definitions but cannot start a proof. Is that normal?
Very. Starting is a separate skill: unpacking the definitions of what you are given and what you must show, trying small examples and looking for the connection. We practice this explicitly, and it usually improves within a few weeks.
Can you help me prepare for a qualifying or comprehensive exam?
Yes, with study planning, review of core theorems and their proofs, and practice on past-style problems under timed conditions, with feedback on your written solutions.
Do you teach LaTeX for writing proofs?
Yes, as part of sessions: theorem and proof environments, aligned equations, and common notation. For larger documents, see our LaTeX and Overleaf support.
Is this different from general math tutoring?
Yes. General math tutoring focuses on school and early university content such as algebra and calculus techniques. This service focuses on proof-based courses and advanced problem solving, where the main difficulty is rigor and mathematical writing.