Second Test:

 


1) Monotone  Convergence (Th2.19) know proof, Nested Interval Property(Th2.23) 
   Bolzano- Weierstrass theorem (Theorem 2.26) 
  

2) Definition of Cauchy sequence. Theorem of Cauchy (Th2.29) know proof.
   Find a Cauchy sequence in Q which does not converge in Q
   (thus completeness is essential for convergence of Cauchy sequences)


3) Limsup and Liminf of sequences. Important Theorems 2.35 and 2.37

4)  Definition for limits of function (Definition 3.1). Prove the five
    limit statements on top of page  62  only using definition 
    (i.e. without using description of limits by sequences).
     One of them will be on test
    Squeeze Theorem Comparison.
5) Sequential Characterization of limits (Th3.6) know proof. Prove all limit theorems
   using that characterization and corresponding limit theorems for
   sequences.

6)  Definition of one sided limits (3.12) and Theorem 3.14  

7) Continuity, Definition 3.19, and sequential characterization (Th 3.21)

8) Very important theorems : Extreme Value Theorem 3.26 and 
   Intermediate Value theorem 3.29. Know the proofs and examples
   which show that these theorems are wrong if some of the assumptions
   are not satisfied.