Foundations

  • Axioms & Definitions
  • Proof by contradiction & Proofs as games & Induction proper
  • Circularity in mathematics & 'Paradoxes' resolved
  • Building blocks & Incompleteness theorems
  • Logic books

Natural deduction

  • Rules & Examples

Rigour with intuition

  • Reals & Real exponentiation & $\cos,\sin$ and $\exp(iπ)+1=0$
  • General differentiation & Its applications
  • Ordinals and cardinals

Beautiful stuff

Puzzles:

  • Rubik's cube & other permutation puzzles

Finite differences:

  • Indefinite summation (like the sum of the first n cubes)
  • Recurrence relations & Indefinite sum of such sequences

Asymptotic expansions:

  • For k-times-differentiable functions
    • Central difference formulae
    • Newton-Raphson with precise convergence criteria
  • Systematic computation of limits

Smoothing:

  • To prove AM-GM & Jensen's inequality
  • Maximum area of polygon given perimeter
  • Maximum expression value using brackets

Multi-dimensional analysis:

  • Multi-dimensional IVT, which proves:
    • Poincaré–Miranda theorem
    • Ham sandwich theorem [consider all hyperplanes that bisect one piece]

Inversion:

  • Apollonius' Problem

Common oversights

  • Teaching separating variables
  • 'One-step' analysis to find expectation
  • Most explanations of induction
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