MATH 3141 - Schedule

  Day     Topics/events     Sections in text  
  8/29     Overview of continuity, differentiability, and integrability. Pitfalls of exchanging limits.     N/A  
  9/1     Field axioms, countability of rationals, rigorous definition of limit.     1.1, 1.2  
  9/5     Properties of limits, monotone sequence property, completeness.     1.2  
  9/7     Density of rationals (in R), infimum/supremum.     1.2, 1.3  
  9/12     Cauchy sequences.     1.4  
  9/14     Cluster points, limsup and liminf.     1.5  
  9/19     (n-dimensional) Euclidean space, norms, metrics, inner-products. Cauchy-Schwarz inequality.     1.6, 1.7  
  9/21     General norm, metric, and inner-product spaces. Open sets.     1.7, 2.1  
  9/26     More on open sets, definition of a closed set.     2.1, 2.3  
  9/28     Accumulation points and interior points, interior and closure of a set.     2.2, 2.4, 2.5  
  10/3     Boundaries, limits and closures.     2.6, 2.7, 2.8  
  10/5     Series, Cauchy criteria, comparison theorems for series.     2.9  
  10/10     Root and rato tests (their successes and failures).     2.9  
  10/12     Test #1.     1.1-1.7, 2.1-2.6  
  10/17     Summation by parts, Alternating series.     N/A (bonus material! Though bits in 2.9)  
  10/19     Rearrangements of series.     N/A (more bonus material!)  
  10/24     Compactness: definition and Bolzano-Weierstrauss.     3.1  
  10/26     Compactness: Heine-Borel, total boundedness, nested set property.     3.2, 3.3  
  10/31     Perfect sets and Cantor sets.     N/A (but see the exercises for Ch. 3)  
  11/2     Continuity, images/pre-images of open, closed, and compact sets.     4.1, 4.2, 4.3  
  11/7     Boundedess and extremes of continuous functions on compact sets, uniform continuity.     4.4, 4.6  
  11/9     Contunuity and connectedness, intermediate value theorem.     3.4, 3.5, 4.5  
  11/14     Test #2.     2.7-4.1, 4.3