Slides & board photos, material covered

Lecture notes: (WARNING: These can change day to day.)

Board photos & synopsis of material covered:

lecture 1, Jan 9: (done by Haiyu Huang) Introduction to propositional logic. Propositions and operations on them. Quantifiers.
lecture 2, Jan 10: Intro to the course. Theory building in mathematics. Review of formalism of propositional logic. Transliteration of English sentences into formal notation. Quantifiers, their negation and stacking.
lecture 3, Jan 11: Proof by contrapositive and contradiction. Set theory: all objects are sets with operation "belongs to". Comprehension principle in naive set theory. Russell's paradox and need for better theory.
lecture 4, Jan 13: Zermelo's axioms of Separation, Extensionality, Pairset, Emptyset, Unionset, Powerset and Infinite set. Cartesian product of two sets. Relation. Reflexivity, antisymmetry, transience and partial order. Equivalence relation. Class of equivalence. A function, its domain and range. Image and preimage of a set under a function.
lecture 5, Jan 17: Inverse relation and inverse function. Injective, surjective and bijective functions. Equinumerosity of sets (example of a bijection) and notion of being Dedekind infinite (example of an injection). Collection of sets indexed by a set. Cartesian product of arbitrary collections of sets. Axiom of Choice. Peano axioms for systems of naturals. Proof by induction.
lecture 6, Jan 18: An intermezzo on Axiom of Choice. Existence of a system of naturals. Recursive definition/construction/principle using a system of naturals. Proof to be done next time.
lecture 7, Jan 20: Statement and detailed proof of the recursive principle. Uniqueness of the system of naturals up to an isomorphism.
lecture 8, Jan 23: Addition defined using recursive principle. Commutativity and associativity of addition, proved by induction. Ordering of naturals. Total ordering. Multiplication and its properties.
lecture 9, Jan 25: Powers and factorial on naturals. Integers as equivalence classes of pairs of naturals. Operations of integers. Rationals as equivalence classes of pairs of integers. Operations on these.
lecture 10, Jan 27: Fields. Uniqueness of the unit elements and inverses. Examples of finite fields. Ordered fields. Rationals form an ordered field. The naturals of an ordered field. An axiomatic definition of a system of rationals. Statement of uniqueness of systems of rationals.
lecture 11, Jan 30: Algebraic deficiencies of rationals. Euclid's proof that no rational squares to 2. Rational Root Test as a tool to verify that a radical is not (or is) rational. Definition of the solution (of a polynomial equation) in radicals. Failure for quintic equation.
lecture 12, Feb 1: Notion of upper/lower bounds in partially ordered sets. Definition of supremum and infimum. Uniqueness of supremum and infimum (if they exist). Examples: supremum and infimum on power sets ordered by set inclusion. (Care needed when dealing with sup(∅) and inf(∅) which are ordered unintuitively!) Infimum of a non-empty set of naturals.
lecture 13, Feb 3: Infimum of a non-empty set of naturals one more time. Validity of Axiom of Choice for collections of such sets. Completeness of an ordered field. Absence of supremum for a bounded set of rationals. Rationals are not complete. A system of reals defined as a complete ordered field.
lecture 14, Feb 6: A model of the reals based on Dedekind cuts. Definition of a Dedekind cut and its properties. Ordering relation on cuts. The ordering being total and complete. Addition and multiplication for Dedekind cuts.
lecture 15, Feb 10: Uniqueness for systems of reals from uniqueness for systems of rationals. Archimedean property of the reals. Density of rationals and irrationals in reals. Existence/uniqueness of a general root of a positive real.
lecture 16, Feb 13: Finite and infinite sets. Cardinality of a finite set. Finiteness preserved under subsets, unions and Cartesian products. Dedekind infinite vs infinite. Countable and uncountable sets. Cartesian product of countable sets. Any subset of a countable set is finite or countable.
lecture 17, Feb 15: Rationals are countable. Countable union of countable sets is countable. Algebraic reals are countable. Examples of uncountable sets: zero-one valued sequences via Cantor's diagonal argument and the reals via embedding. Equinumerosity relation and cardinality defined via equivalence classes thereof. Powerset of a set is not equinumerous to the set itself.
lecture 18, Feb 17: Size comparison and Cantor/Schroeder-Bernstein theorem. The continuum and beyond. Sequences of real numbers. The property of being Cauchy (or clustering) and having a limit, or being convergent. Proof of convergence of a recursively defined sequence. Notion of a metric space.
lecture 19, Feb 22: Two metrics on the reals. Euclidean metrics, box metric and Manhattan metric on "Euclidean" spaces. A p-norm and the corresponding metric. Discrete metric. Cauchy and convergent sequences in metric space. Uniqueness of the limit. Convergent implies Cauchy. Convergence and Cauchy in discrete metric equivalent to being eventually constant. Subsequence and characterization of convergence by being Cauchy and having a convergent subsequence.
lecture 20, Feb 24: Open ball. Open and closed sets. Singletons are closed while open balls are open. All sets in discrete metric are open and closed. The set of open sets in a metric spaces is closed under arbitrary unions and finite intersections. The notion of topology (a.k.a. abstract open sets). Closed sets, interior, closure, topological boundary. Characterization of open subsets of real line.
lecture 21, March 6: Adherent, interior, exterior, boundary, isolated and limit points and their characterization using concepts from topology. Sequential characterization of closed sets in a metric space. A similar statement for the closure. Dense sets.
lecture 22, March 7: Relative topology via induced metric. Complete metric space. Completeness of the reals under Euclidean metric. Bounded monotone sequences converge. The Bolzano-Weierstrass theorem. Completeness of ℝ^d with respect to all norm metrics. Comparison of norms in ℝ^d, equivalence of metric convergence with convergence of components as reals.
lecture 23, March 8: Inheritance of completeness to closed subsets and Cartesian products. The notion of a contraction map. Banach's contraction principle. Completion of a metric space; its uniqueness modulo an isometry and existence. (Details relegated to the notes.)
lecture 24, March 10: Sequential compactness. Necessary conditions: boundedness, closedness and (for spaces) completeness. The Heine-Borel property (boundedness and closedness imply sequential compactness) for Euclidean spaces. Failure of this in infinite dimension.
lecture 25, March 13: Total boundedness and its sufficiency, along with completeness, for sequential compactness. First Cantor's proof of uncountability of reals. Cantor's intersection property.
lecture 26, Feb 14: Equivalent formulation using countable open cover property. Compactness via arbitrary open covers. Separability of sequentially compact spaces. Reduction of arbitrary open covers to countable covers for separable spaces (Lindelöf's lemma). Extended reals, supremum and infimum, algebraic operations with infinity symbols. Limsup and liminf. Equality of these is equivalent to convergence for bounded sequences.
lecture 27, Mar 15: Improper limits. Operations with limits. Infinite series as the limit of sequence of partial sums. Geometric series and expansion of reals in decimals. Sufficient conditions for convergence. Divergence of the harmonic series. Convergence under any faster polynomial decay.
lecture 28, Mar 17: Cauchy criterion and characterization of convergence by vanishing limit of the (sequence of) tails. Absolutely convergent series. Absolute convergence implies convergence and summability in any order. Conditional convergence and Riemann's rearrangement theorem. Read notes for Cauchy product and criteria for absolute convergence: ratio test (with limsup and limit), root test.