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user21820
Foundations
- Axioms & Definitions
- Proof by contradiction & Proofs as games & Induction proper
- Circularity in mathematics & 'Paradoxes' resolved
- Building blocks & Incompleteness theorems
- Logic books
Natural deduction
Rigour with intuition
- Reals & Real exponentiation & $\cos,\sin$ and $\exp(iπ)+1=0$
- General differentiation & Its applications
- Ordinals and cardinals
Beautiful stuff
Puzzles:
Finite differences:
- Indefinite summation (like the sum of the first n cubes)
- Recurrence relations & Indefinite sum of such sequences
Asymptotic expansions:
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:
Common oversights
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