**Foundations**

- Axioms & Definitions
- Contradiction & Game semantics & True induction
- Circularity in mathematics & 'Paradoxes' resolved
- Building blocks & Incompleteness theorems
- Logic texts

**Natural deduction**

**Rigour + Intuition**

- Naturals
- Reals & Real exponentiation & $\cos,\sin$ & $\exp(iπ)+1=0$
- General differentiation & applications
- Matrix multiplication

**Beautiful stuff**

Puzzles:

Finite difference:

Asymptotic expansion:

- k-times-differentiable function
- Central difference formulae
- Newton-Raphson 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:

**Common oversights**