Instructor Dragos Oprea
Lectures: MWF 1:00-1:50PM, PETER 102
Daniel Drimbe
  • Discussion 1: APM 7421, Wednesday, 6:00-6:50PM
  • Discussion 2: APM 7421, Wednesday 7:00-7:50PM
  • Office: APM 6414
  • Office hour: Monday 10-11, Wednesday 10-11AM, Thursday 12-1.
  • Email: ddrimbe at ucsd dot edu.

First course in a rigorous, entirely proof-based, three-quarter sequence on real analysis. Topics include: the real number system, basic point-set topology, numerical sequences and series, continuity. Students may not receive credit for both Math 140A and Math 142A. The two courses cover similar topics, but Math 140A does so in more depth and from a more theoretical perspective.


Math 31CH (Honors Vector Calculus) or Math 109 (Mathematical Reasoning), or consent of the instructor. Math 140 is a difficult and time consuming course, so enroll only if your course load allows it.

The grade is computed as the following weighed average:

  • Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.

W. Rudin, Principles of Mathematical Analysis, Third Edition.


Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.


Homework problems will be assigned on the course homework page. There will be 7 problem sets, due Fridays at 4:30PM in the TA's mailbox. You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.


There will be two midterm exams given on October 24 and November 21. There will be no makeup exams.

Regrading policy: graded exams will be handed back in section. Regrading is not possible after the exam leaves the room.


The final examination will be held on Monday, December 15, 11:30-2:30AM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.

Announcements and Dates
  • Friday, October 3: First lecture
  • Friday, October 24: Midterm I
  • Tuesday, November 11: Veterans' Day.
  • Friday, November 21: Midterm II
  • Thursday-Friday, November 27-28: Thanksgiving. No class.
  • Friday, December 12: Last Lecture
  • Monday, December 15: FINAL EXAM, 11:30-2:30.
  • Preparation for Midterm 1:
  • Preparation for Midterm 2:
  • Preparation for Final Exam:
Lecture Summaries
  • Lecture 1: Introduction and motivation. Ordered sets. Lower bound, upper bound. Infimum and supremum.
  • Lecture 2: Fields and their axioms. Examples. Ordered fields. The definition of the field of real numbers.
  • Lecture 3: Comments on the construction of the real numbers. Archimedian property. Density of rational numbers. Existence of nth roots of positive real numbers.
  • Lecture 4: The extended real number system. Complex numbers. Definition and properties.
  • Lecture 5: Cauchy-Schwartz inequality. Euclidean spaces and properties. The triangle inequality. Sets, functions, bijections.
  • Lecture 6: Countable and uncountable sets. Rational numbers are countable. Real numbers are uncountable. Countable union of countable sets is countable. Finite products of countable sets are countable.
  • Lecture 7: Metric spaces and examples. Neighborhoods. Interior points. Open sets and properties.
  • Lecture 8: Limit points. Isolated points. Closure. Closed sets. Dense sets. Properties of closed sets. Supremum belongs to the closure. Open/closed sets of metric subspaces.
  • Lecture 9: Open covers. Compact sets. Compactness is not a relative concept. Compact sets are closed. Closed sets of compact sets are compact. Nonemptyness of intersections of compact sets.
  • Lecture 10: Nested closed and bounded intervals have non-empty intersections. Infinite subsets of compact sets have limit points. Generalizations to k-cells.
  • Lecture 11: k-cells are compact. In R^k, compact sets are closed and bounded and conversely. Sequential compactness and Weierstrass' theorem.
  • Lecture 12: Convergence in metric spaces and its properties. Limits of real sequences and their behaviour with respect to addition and multiplication.
  • Lecture 13: For monotonic sequences, boundedness and convergence are equivalent. Sequantial compactness. Cauchy sequences and convergence. Complete metric spaces.
  • Lecture 14: Diagram connecting the type of sequences we encountered. Special sequences and their limits. Limits at infinity. Liminf and limsup.
  • Lecture 15: Series and their sums. Cauchy criterion. Comparison test. Series with non-negative coefficients.
  • Lecture 16: Geometric series. Cauchy condensation test. Harmonic series.
  • Lecture 17: Two definitions of the number e in terms of limits of series/sequences. Irrationality of e.
  • Lecture 18: Root test and ratio test. Comparison of the two tests. Some examples.
  • Lecture 19: Power series and radius of convergence. Abel summation. Alternating series test.
  • Lecture 20: Absolute convergence. Addition and multiplication of power series. Cauchy product of series. Mertens theorem on products of series.
  • Lecture 21: Rearrangements of series. Absolute convergence and convergence of rearranged series. Limits of functions.
  • Lecture 22: Limits of functions and connection with limits of sequences. Continuity. Properties of continuous functions: sums, products, composition, etc.
  • Lecture 23: Continuity and topology: preimages of open and closed sets are open and closed. Images of compact sets are compact. Weierstrass extreme value theorem. Discussion of the continuity of inverses.
  • Lecture 24: Uniformly continuous functions. Lipschitz continuity implies uniform continuity. Continuous functions over compact sets are uniformly continuous.
  • Lecture 25: Connected sets. Intervals of the real line are connected and conversely. Images of connected sets under continuous maps are connected. Intermediate value property. Darboux functions.
  • Lecture 26: One-sided limits. Discontinuities of first and second kind. Examples. Discontinuities of Darboux functions are of second kind. Discontinuities of monotonic functions are of first kind.
  • Lecture 27: Monotonic functions only have at most countably many discontinuities. Limits at infinity and to infinity.