WEEK 1

triangular numbers, Sigma Notation for Summation, Sequences, Peano's axioms for the natural numbers,
Set Theory: the language of Mathematics, Hilbert Hotel, Bijections and Cardinality, The Natural Numbers,
The Integers

WEEK 2

A Trip to Cantorsville, Cantor's Diagonalization Argument, Towards the Real Numbers, Ordered Field,
Completeness Axiom, The Least Upper Bound Property, Mathematical Logic and Statements

WEEK 3

Currency Game, Divisibility, Greatest Common Divisor, The Euclidean Algorithm, Proof of the Euclidean
Algorithm, Test for Divisibility

WEEK 4

Fermat's Little Theorem, Fundamental Theorem of Arithmetic, Modular Arithmetic, Arithmetic with
Congruences, Infinitude of Primes, Inclusion–Exclusion Principle

WEEK 5

Proof of InclusionExclusion Principle, Pigeonhole Principle, Sieve of Eratosthenes, More on the
Infinitude of Primes, Gaps Between Primes

WEEK 6

Binomial Coefficients, Binomial Theorem, Lattice Paths, Random Sampling, Permutations, Example
of InclusionExclusion Principle

WEEK 7

Graph Theory  Introduction, Connectedness and Distance in Graphs, Adjacency Matrix, Graph Trees,
Sperners Lemma, Graph Coloring

WEEK 8

Limit of a Sequence, Properties of Limits, Sequential Continuity, Continuity of Trigonometric
functions, Intermediate Value Theorem

WEEK 9

Limit of Fibonacci series, Lamé formula, Sandwich Lemma, Shifting Lemma

WEEK 10

Derivatives, Faà di Bruno's formula

WEEK 11

Riemann Integrals, Mean Value Theorem

WEEK 12

Uniform continuity, Fundamental theorem of calculus
