Math 409300  Test 1 Review  Summer 2015
General Information
Test 1 will be given on Friday, 7/3/15, during our usual class
time and in our usual classroom. I will have extra office hours on
Thursday afternoon, 14 pm, and on Wednesday morning, 9:3010:30 am.
 Bluebooks. Please bring an 8½x11 bluebook.

Calculators. You may use scientific calculators to do numerical
calculations — logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
 Structure and Coverage The test will have 6 to 8
questions, some with multiple parts. It will cover chapters 1, 2 and
3, except for sections 1.5, 1.6, and 3.4. You should expect to
have to work problems similar to homework problems or problems done in
class. Be able to state definitions or theorems from the first list,
and be able to prove the theorems in the second list.
 Keys to Test 1 and three quizzes from Summer 2005
Definitions and Statements of Theorems
You are expected to know definitions for, or be able to state, the
following:

 Real numbers
 Triangle inequalities
 Completeness axiom
 Bounded set, supremum, infimum
 Approximation Property for Suprema, Infima
 Wellordering principle
 Sequences
 Sequence
 Subsequence
 Limit of a sequence
 Bounded sequences
 Monotone (increasing, decreasing) sequences
 Monotone Convergence Theorem (for sequences)
 Nested sequence of sets
 Nested Interval Theorem
 BolzanoWeierstrass Theorem
 Cauchy sequence
 Theorem (Cauchy)  Every Cauchy sequence is convergent
 Functions of a continuous variable
 Limit of f(x) as x → a; onesided limits; limits as x →
∞
 Sequential characterization of limits
 Continuous function on a set E
 Sequential characterization of continuity
 Composition of g with f, g∘f. Know that
If f:A → B, g:B→
R, and both f and g are continuous on A and B, respectively,
then g∘f is continuous.
 Bounded function
 Extreme Value Theorem
 Intermediate Value Theorem
Proofs
Be able to prove the following:
 Archimedean principle
 Approximation Property for Suprema, Infima
 Monotone Convergence Theorem for Sequences
 BolzanoWeierstrass Theorem
 Extreme Value Theorem
Updated 6/30/2015 (fjn)