Here is a list of topics to be covered in Midterm I. You are responsible for all material 
from lectures, homework, and the posted notes. 


Chapter 1: 

 -- natural numbers (Peano axioms); mathematical induction. 
(you will not be asked to state the axioms, but you must know how to use the method of induction.)
practice questions 1.5, 1.8, 1.9 on p. 5


-- rational, irrational, algebraic numbers. 
(1. you should be able to show, in simple cases, that a given number is algebraic -- just 
build the required polynomial; 2. show, in simple cases, that a given number is irrational --
use rational zeroes theorem) 
practice: Example 1 p.8, questions 2.1, 2.6, 2.7 on p.13

-- Upper bound, lower bound, supremum, infimum of a set. (know the definitions, be able 
to find bounds in simple cases.) 
practice: 4.1-4.6, p.27

-- Completeness Axiom (memorize the statement, understand what it means)

-- Archimedean Property (memorize the statement; understand why it's important and how it follows 
from Completeness Axiom). Denseness of rational numbers (know the statement and the idea of proof).
practice: 4.10, 4.11

***you are not required to memorize any of the algebraic or order axioms from section 3***.

Chapter 2: 

-- Sequences (understand difference between a sequence and a set)
practice: example 1 p.33 

-- Limits: neighborhoods and tails, epsilon definition (be able to write the definition, 
    understand its meaning and various restatements). Practice: 7.4
-- Working with specific sequences: finding limits and proving your answer directly 
   from definition, using limit laws (both finite limits and limits equal to +infinity or -infinity)
   Be able to show that a given sequence diverges (subsequences are often useful for that). 
Practice: 8.1, 8.2, 9.1, 9.2, 9.8


-- Properties of convergent sequences: convergent sequence is bounded, uniqueness of limits 
   (know and understand proofs). Be able to prove various other properties working from the 
   definition of the limit. 
Practice: 8.4, 8.9, 8.10

-- Proving Limit Laws (and their versions for infinite limits, when applicable). You should
   also be able to prove statements like the one given in the quiz.  
Practice: 8.3, 9.9-9.11 + make your own questions like the quiz


-- Theorem: monotonic bounded sequence converges (know proof). what happens when the sequence
   is not  bounded?  Use this theorem to find limits of sequences (often defined by recurrent relations)
Practice: 10.2, 10.5; 10.9, 10.10
    
-- subsequences: definition, finding subsequences with certain properties. 
   Theorem: if a sequence converges, all its subsequences converge to the same limit.
   (this is useful for proving divergence!)
Practice: 11.5, 11.6, 11.7; Theorem 11.2 p.68


-- Bolzano-Weierstrass theorem: every bounded sequence has a convergent subsequence. 
   (memorize the statement, understand what it means. what happens if the sequence is not bounded?)


-- Series (only series with positive terms as covered in class). See posted notes.  
Practice: 14.5, 14.7, 14.9

**** Topics not covered in class and not detailed here are not on the test. In particular, 
Cauchy sequences (section 10) and notions of lim inf and lim sup (section 11) are not required. ****