# Math 409-300 - 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, 1-4 pm, and on Wednesday morning, 9:30-10:30 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
Well-ordering 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
Bolzano-Weierstrass Theorem
Cauchy sequence
Theorem (Cauchy) - Every Cauchy sequence is convergent

Functions of a continuous variable
Limit of f(x) as x → a; one-sided 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
• Bolzano-Weierstrass Theorem
• Extreme Value Theorem

Updated 6/30/2015 (fjn)