Proof writing tips.

Lesson 1 Ch1 The ordered field (R,+,*,<) ; Definition of Sup and inf; completeness axiom.

Lesson 2 Ch1 Definition of inf; examples;Properties of sup and inf;

Lesson 3 Ch 1 square root of 2; Q is dense in R

Lesson 4 Ch 2 Useful identities/inequalities. Review of Induction Definition of limit of a sequence Examples

Lesson 5 Ch 2 Example of limits , limit laws, squeeze theorem

Lesson 6 Ch 2 Proofs of limit theorems. Some other general theorems

Lesson 7 Ch 2 Proof of limit theorem #4. Sequences divergent to infinity

Lesson 8 Ch 2 Monotone sequences. Subsequences

Lesson 9 Ch 2 Subsequence theorem. Sets dense in R . Open and closed sets

Lesson 10 Ch 2 Open and closed sets. Sequentially compact sets . The number e (optional)

Lesson 11 Ch 9 Cauchy sequences. Series definition. Geometric series

Lesson 12 Ch 9 Series . Geometric series. Harmonic series

Next Monday will be a review day. No lesson will be posted for Monday . We may do some problems from this exam Beykel Win 2016

Lesson 13 Ch 9 Comparison test. Limit test

Lesson 14 Ch 9 Alternating series test. Absolute convergence

Lesson 15 Ch 9 Ratio test

Lesson 16 Ch 3 Limit of a function. Limit laws. Limit of composition

Lesson 17 Ch 3 Continuity. Examples. Sum, product, quotient of continuous functions.

Lesson 18 Ch 3 Continuity of composition. More examples.

Lesson 19 Ch 3 Extreme value theorem.

Proof review problems we did not do on Monday in class

Lesson 20 Ch 3 Intermediate value theorem.

Lesson 21 Ch 3 Uniform continuity.

Lesson 22 Ch 3 Monotonic functions. Inverses

Lesson 23 Ch 9 Sequences of functions. Pointwise and Uniform convergence

Lesson 24 Ch 9 Sequences of functions. Examples