Lectures for 18.100B, Section 1, Spring 2007 -- actual content.

1. Lecture 1
   • Field axioms
   • Ordered sets
   • Reals and rationals are ordered fields.
   • Least upper bound property of the reals
   • There is no rational with square 2.
2. Lecture 2
   • Sets and maps, 1-1, onto
   • Cardinality
   • Countability of a countable union of countable sets.
   • Uncountability of set of sequences with values 0,1.
   • Archimedian principle.
3. Lecture 3
   • Metric spaces
   • Distance function, discrete metric
   • Real line, triangle inequality
   • Complex numbers form a field
   • Euclidean spaces -- Schwarz' inequality.
   • Open balls, open sets.
   • Theorem: open balls are open.
4. Lecture 4
   • Arbitrary unions and finite intersections of open sets are open
   • Limit points
   • Closed sets
   • A set is closed iff its complement is open.
   • Arbitrary intersections and finite unions of closed sets are closed.
5. Lecture 5
   • Closure of a set -- union with limit points.
   • Closure as intersection over all closed supersets.
   • Relatively open sets
   • Open covers, subcovers and compact sets.
   • Compact sets are closed.
   • Closed subsets of compact sets are compact.
6. Lecture 6
   • Intersection property of compact sets
   • Decreasing sequences of compact sets
   • Limit points of infinite subsets of compact sets
   • Compact subsets of Euclidean space (proof not quite finished).
7. Lecture 7
   • Heine-Borel completed
   • Weierstrass' theorem
   • Connected sets in a metric space
   • Connected subset of the real line
   • Sequences and convergence
   • Complex sequences
8. Lecture 8
   • Sequences of complex numbers
   • Sequences in Euclidean space
   • Subsequences
   • A sequence in a compact space has a convergent subsequence
   • The subsequential limits form a closed set
   • Cauchy sequences
9. Lecture 9
   • Diameter of a set
   • Cauchy sequences in a compact space converge
   • Cauchy sequences in Euclidean space converge
   • Completeness
   • Monotonic sequences or real numbers
   • Extended real numbers, sup and inf
   • Extended set of subsequential limits of a real sequence
10. Lecture 10
    • Examples of sequences
    • Series, Cauchy criterion
    • Comparison test
    • Serries with decreasing positive terms.
    • e
11. Lecture 11
    • Irrationality of e
    • Root test
    • Ratio test
    • Absolute convergence
    • Power series, radius of convergence
12. Lecture 12
    • Maps and functions
    • Limit of a map at a limit point
    • Sequences and limits
    • Sums and products of functions
    • Continuity at a point
    • Composite functions
    • Continuity and open sets (begun)
13. Lecture 13
    • Distance function is continuous
    • Uniform continuity
    • Continuous map from a compact set is uniformly continuous
    • The continuous image of a compact set is compact
    • Continuous functions on compact sets acheive their maximum and minimum
    • A continuous bijection from a compact set has continuous inverse
    • The continuous image of a connected set is connected
14. Lecture 14
    • Continuous image of connected set is connected (completed)
    • Differentiablility
    • Differentiability at a point implies continuity
    • Sums, products and quotients
    • Composite functions
    • Derivative vanishes at a local max or min.
15. Lecture 15
    • Mean value theorem
    • Intermediate value theorem for derivatives
    • L'Hopital's rule
    • Higher order derivatives
    • Taylor's theorem (not quite finished)
16. Lecture 16
    • Taylor's theorem
    • Partitions and upper and lower sums
    • Refinement
    • Riemann-Stieltjes integrability
17. Lecture 17
    A bit grueling!
    • Continuous functions are R-S integrable
    • Montonic functions are R-S integrable against continuous alphas
    • Continuous except at finite set is integrable if alpha is continuous there.
    • Continuous function of an R-S integrable function is R-S integrable.
18. Lecture 18
    • Additivity and other properties of the integral
    • Differentiable alpha
    • Integral continuous and differentiable where integrand continuous
    • Fundamental theorem of calculus
    • Change of variable
19. • Integration by parts
    • Uniform convergence of sequences of functions
    • Uniform limit of continuous functions is continuous
    • Uniform Cauchy sequence is uniformly convergent
20. • Completeness of space of continous functions
    • Uniform limit of integrable functions is integrable
    • Uniform convergence of derivatives
21. • Existence of continuous, nowhere differentiable function
    • Equicontinuous families
    • Theorem of Ascoli-Arzela.
22. • Power series
    • Infinite differentiability of analytic functions
    • Exponential function
    • Logarithm
23. • Trigonometric functions
    • Fundamental theorem of algebra
    • Winding number
    • FTA again
24. • Riemann versus Lebesgue integration.
    • Which course to take next?
    • Problems as requested.
    • Evaluation forms.
25. • Revision -- once through the course lightly!