Real Analysis

Calculus made rigorous: the completeness of ℝ and the supremum, the ε–N limit, sequences and series, compactness, and the Riemann integral.

Foundations of Analysis

Supremum & Completeness — Least upper bounds, infimum, and the axiom that separates the reals from the rationals.
Nested Interval Property — Nested closed bounded intervals trap a single real number.
Open & Closed Sets — Interior points, closure, and why closed is not just not open.

ε-N Convergence — Making "gets close to" precise: for every tolerance, a threshold that works.
Monotone Convergence Theorem — A bounded monotone sequence converges to its supremum or infimum.
Cauchy Sequences — Terms that crowd together — and why that means convergence in the reals.
Limit Superior & Inferior — An eventual ceiling and floor that exist even with no limit.
Series Convergence Tests — The nth-term, p-series, and ratio tests.
Radius of Convergence — The disk where a power series converges, and its boundary.

Bolzano–Weierstrass Theorem — Every bounded real sequence has a convergent subsequence.
Compact Sets (Heine–Borel) — Compact means closed and bounded — every open cover has a finite subcover.
Uniform Continuity — One δ for the whole domain — a global strengthening of continuity.
Uniform Convergence — Pointwise vs uniform: when the limit of nice functions stays nice.
Riemann Integrability — Darboux upper and lower sums, and when they define an integral.