These are all elementary;
other core math is more technical (manifolds, groups, analysis)
number theory:
sqrt(2) (-> alg ints, p-adics)
infinitely many primes (contradict/construct)
sum of 2 squares (-> mod 4, Gaussian integers)
set theory:
infinite (Archimedean: sand counter; arbitrarily large; naming)
countability (and infinite, cofinite; infinite, coinfinite; Z, 2Z, Q all
same size)
uncountability (Cantor's diagonalization argument (general technique,
like induction))
(do Russel's paradox concretely for maps 3 -> P(3): see that don't get
barber set;
"working simple examples" is general technique for understanding
proofs,
rather than always only working via deduction only)
combinatorics:
binomial coefficients / Pascal's triangle