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Courses Mathematical Foundations for Quant Finance Proof Gym

Proof Gym

Proofs live here, not in the lessons. Weekly practice in four formats (reorder the steps, fill the gap, find the error, write it yourself), so a proof-based course feels like a step, not a cliff.

  • 6 stages
  • 32 lessons
  • About 19 hours
  • Alongside Levels 0–5

Six stages

  1. 1PG1 · runs with Levels 0–1Logic and language
  2. 2PG2 · runs with Levels 1–2Proof techniques
  3. 3PG3 · runs with Levels 1–2Induction
  4. 4PG4 · runs with Level 2ε–δ limits
  5. 5PG5 · runs with Level 2Series and convergence
  6. 6PG6 · runs with Levels 3–5Sets and functions

PG1 · Logic and language

Statements, Quantifiers, Implication, Negation, For every ε there exists δ. Worked example: Negate "the fund beats the index every year" correctly.0 of 5 done · about 2½ hours
  1. PG1.1Statements and truth valuesSimulate, then reorder the steps and find the error · 25 minStart →
  2. PG1.2Implication, converse and contrapositiveSimulate, then reorder the steps, find the error and write it · 30 min
  3. PG1.3Quantifiers: for every and there existsSimulate, then fill the gap and find the error · 30 min
  4. PG1.4Negating a statementSimulate, then reorder the steps, find the error and write it · 30 min
  5. PG1.5Stage review: for every ε there exists δSimulate, then write it · 35 min

PG2 · Proof techniques

Direct proof, Contrapositive, Contradiction, Cases. Worked example: √2 is irrational; the exponent and log rules from Level 1, proved.0 of 6 done · about 3½ hours
  1. PG2.1Direct proof: even and odd numbersSimulate, then all four formats · 30 min
  2. PG2.2The exponent rules, provedSimulate, then reorder the steps, fill the gap and find the error · 35 min
  3. PG2.3The logarithm rules, provedSimulate, then fill the gap, find the error and write it · 35 min
  4. PG2.4The contrapositive, and proof by casesSimulate, then reorder the steps, fill the gap and find the error · 35 min
  5. PG2.5Proof by contradiction: √2 is irrationalSimulate, then reorder the steps, find the error and write it · 35 min
  6. PG2.6Stage review: choosing a techniqueSimulate, then write it · 40 min

PG3 · Induction

Weak induction, Strong induction. Worked example: The geometric-series and annuity formulas.0 of 5 done · about 3 hours
  1. PG3.1Why induction worksSimulate, then reorder the steps · 30 min
  2. PG3.2The geometric series, provedSimulate, then all four formats · 35 min
  3. PG3.3The annuity formula, provedSimulate, then all four formats · 35 min
  4. PG3.4Strong inductionSimulate, then fill the gap and find the error · 40 min
  5. PG3.5Stage review: two proofs of your ownSimulate, then write it · 40 min

PG4 · ε–δ limits

ε–N for sequences, ε–δ for functions, Sup and inf, Completeness. Worked example: (1 + 1/n)ⁿ converges.0 of 6 done · about 4 hours
  1. PG4.1What a limit means: ε and NSimulate, then reorder the steps, fill the gap and find the error · 40 min
  2. PG4.2Limits of sums and productsSimulate, then all four formats · 40 min
  3. PG4.3Limits of functions: ε and δSimulate, then reorder the steps, fill the gap and find the error · 40 min
  4. PG4.4Supremum, infimum and completenessSimulate, then fill the gap, find the error and write it · 40 min
  5. PG4.5Monotone sequences, and why (1 + 1/n)ⁿ convergesSimulate, then reorder the steps, fill the gap and find the error · 40 min
  6. PG4.6Stage review: limits from the definitionSimulate, then write it · 40 min

PG5 · Series and convergence

Convergence tests, Absolute convergence, Uniform convergence (light). Worked example: The Taylor series of eˣ converges for every x.0 of 5 done · about 3½ hours
  1. PG5.1A series is the limit of its partial sumsSimulate, then reorder the steps, fill the gap and find the error · 40 min
  2. PG5.2The comparison and ratio tests, provedSimulate, then all four formats · 45 min
  3. PG5.3Absolute and conditional convergenceSimulate, then fill the gap, find the error and write it · 40 min
  4. PG5.4The series for eˣ converges for every xSimulate, then reorder the steps, fill the gap and find the error · 45 min
  5. PG5.5Stage review: proving a series convergesSimulate, then write it · 40 min

PG6 · Sets and functions

Set operations, Injective and surjective maps, Countable and uncountable sets. Worked example: Why infinitely many coin tosses needs measure theory.0 of 5 done · about 2½ hours
  1. PG6.1Sets and set operations, provedSimulate, then reorder the steps, fill the gap and find the error · 30 min
  2. PG6.2Injective, surjective and bijective mapsSimulate, then reorder the steps, find the error and write it · 30 min
  3. PG6.3Countable setsSimulate, then reorder the steps, fill the gap and find the error · 30 min
  4. PG6.4Uncountable sets, and infinitely many coin tossesSimulate, then reorder the steps and find the error · 30 min
  5. PG6.5Stage review: sets, maps and sizesSimulate, then write it · 30 min

Every proof starts with the numbers

A lesson first checks the claim in Python, then states it, then proves it in up to four formats, easiest first.

  • 1. Reorder the steps

    The lines of a proof, shuffled. Put them back in order, with the arrows or by dragging.

  • 2. Fill the gap

    A proof with its key steps missing. Supply each one, checked like a quick answer.

  • 3. Find the error

    A proof that looks right but is not. Pick the faulty line and say what is wrong.

  • 4. Write it

    Write the proof yourself, then compare it with a model answer and a checklist.