Math 409 - Test I Review
General Information
Test 1 (Friday, 7/1/05) will have 6 to 8 questions, some
with multiple parts. It will cover chapters 1, 2 and 3, except for
sections 1.4 and 2.5. The format will be similar to the quizzes you've
taken. You should expect to have to state definitions - and some
theorems -, work problems similar to ones on the homework and quizzes,
and be able to prove theorems.
Definitions and Statements of Theorems
You are expected to know definitions for, or be able to state, the
following:
- Real numbers
- well-ordering principle
- completeness axiom
- bounded set, supremum, infimum
- Approximation Property for Suprema, Infima
- Sequences
- sequence
- limit of a sequence
- bounded sequences
- monotone (increasing, decreasing) sequences
- Monotone Convergence Theorem
- nested sequence of sets
- Nested Interval Theorem
- Bolzano-Weierstrass Theorem
- Cauchy sequence
- Cauchy's Theorem - 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
- bounded function
- Extreme Value Theorem
- Sign-Preserving Lemma
- Intermediate Value Theorem
- uniform continuity
- Uniform Continuity Theorem - p.~81
Proofs
Be able to prove the following:
- Approximation Property for Suprema, Infima
- Monotone Convergence Theorem for Sequences
- Sequential Characterization of (Continuous) Limits
- Extreme Value Theorem
- Uniform Continuity Theorem (Theorem 3.39 in the text.)
Problems
Be familiar enough with the following items to be able to use them to
solve problems similar to ones on quizzes and homework: notation for
various sets, limits etc.; mathematical induction; binomial theorem;
density of rationals or irrationals among reals; various limit
theorems (these apply to all cases) - squeeze theorems, comparison
theorems, algebraic operations with limits; limits with ∞ as an
answer.