18.100A: Real Analysis

This is the course website for the course 18.100A Spring 2017 with material and information relevant to the course.
The class meets 11:00-12:00h on MWF at 4-163.

Content: Syllabus of the course, with calendar.

Textbook: Introduction to Real Analysis, by A. Mattuck.

Grade scheme: 1/3·PSets +1/3·Midterms + 1/3·Final.

Stellar site: 18.100A LMOD.

Problem Sets
Problem Set 1, due to Friday Feb 17.

Problem Set 2, due to Friday Feb 24.

Problem Set 3, due to Friday Mar 3.

Midterm 1, Mar 6.

Problem Set 4, due to Monday Mar 20.

Problem Set 5, due to Monday Apr 3.

Problem Set 6, due to Friday Apr 14.

Midterm 2, Apr 19.

Problem Set 7, due to Friday May 5.

Lecture Diaries
Part I: Real Sequences

1. Feb 8: Introduction to 18.100A.

The real numbers. Decimal approximations.
Sequences of real numbers. Limit of a sequence.
Bounded monotone sequences converge. Examples
Exponential sequences.

2. Feb 10: Bounds. Limits: Epsilon definition.

Bounding fractions. Bounds with integrals.
Estimations and approximations. n large enough.
Epsilon definition of a limit. Examples.

3. Feb 15: Kε Principle. Error terms.

Computation of limits using the ε-definition.
The Kε Principle. The limit of the sequence an= αn.
The Error Theorem. Estimation of the geometric sequence.
Error bounds and convergence to the square root of two.

4. Feb 17: Limit Theorems. Squeezing. Subsequences.

Properties of Limits. Linearity, product, quotients.
The Squeezing Theorem. Stirling estimate for ln(n!).
Convergent sequences have convergent subsequences.

5. Feb 21: Nested Intervals. Bisection.

Nested intervals. Motivation with ln(2).
Proof of the Nested Intervals Theorem.
Method of Bisection. Examples.
Reals as limits of a rational sequences.

6. Feb 22: Cluster points. Bolzano-Weierstrass.

Cluster points. Five examples with different cluster sets.
Equivalence of cluster points and limits of subsequences.
Bolzano-Weierstrass: Bounded sequences have cluster points.

7. Feb 24: Cauchy sequence. Cauchy criterion.

Cauchy sequences. Examples and non-examples.
Equivalence of Cauchy condition and convergence.
Sum of the square inverses is Cauchy. (Telescoping trick.)
Fibonacci fractions. Contractive sequences converge.

8. Feb 27: Partial sums. Infinite series.

Partial sums and infinite series. Examples.
From sequences to series. Telescoping series.
Necessary conditions for convergence. Fast decay.
Absolute convergence. Comparison criterion.

9. Mar 1: Convergence Tests.

Ratio test. Root test. Non-Examples.
Integral and asymptotic comparison test.
Cauchy's test for alternating series.

10. Mar 3: Power series.

Definition of power series. Examples.
Radius of convergence. Interval of convergence.
Behaviour at boundary points. Geometric series.

11. Part II: Real Functions

12. Mar 8: Continuity of Real Functions.

Local and global properties of functions.
Continuity: the epsilon-delta definitions.
Examples of local properties implying global.

13. Mar 10: Limits of functions. Sequential Continuity.

Limit of a function. Left-right limits.
Classification of discontinuities. Examples.
Sequential characterization of continuity.

14. Mar 13: Bolzano and Intermediate Value Theorem.

Bolzano's Theorem. Intermediate Value Theorem.
Applications to root finding and estimate counts.
Intermediate Value Property and continuity.
Inverse Function Theorem.

15. Mar 15: Local-to-global Theorems.

Local properties. Sequential compactness.
Compact set is equivalent to closed and bounded.
Continuous function on a compact interval is bounded.
Continuous function on a compact interval has a maximum.

16. Mar 17: Uniform continuity. Derivative of a function.

Dependence of delta on epsilon and the point.
Definition of uniform continuity. Non-Examples.
Continuous on compact interval are uniformly continuous.
Derivative of a function at a point. Algebraic Rules.

17. Apr 3: Integrable functions.

Partitions. Lower and upper sums.
Definitions of integrable function.
Monotone functions are integrable. Examples.
Continuous functions are integrable. Non-examples.
Linear combinations preserve integrability.

18. Apr 5: The Riemann Integral.

Partitions refinements and sums.
Existence of the Riemann Integral. Examples.
Riemann sums of integrable functions. Properties.
Interval addition for the Riemann Integral.