MATH 350, Review for Midterm Test 1

Test topics

1. The real number system
   1. Sets and functions (1.1)
   2. Properties of the real numbers as an ordered field (1.2)
   3. Supremum and infimum of a set. The completeness axiom (1.3)
   4. Cardinality. Countable and uncountable sets (1.3)

2. Sequences of real numbers
   1. Sequences and limits (2.1)
   2. Subsequences, subsequential limits (2.2)
   3. Limit points of sets of real numbers; the Bolzano-Weierstrass principle (2.3)
   4. The upper and lower limits (2.3)
   5. Cauchy sequences (2.3)

Important definitions/facts

• Functions; direct and inverse images
• One-to-one, onto and invertible functions
• Axioms of a field, axioms of order; the completeness axiom
• Bounded sets; upper and lower bounds; supremum and infimum
• The Archimedian principle
• Cardinality, countable sets
• Sequence; limit of a sequence
• Subsequences; subsequential limits
• Monotone increasing and decreasing sequences
• Cauchy sequences
• Limit point of a set (a.k.a. cluster, or accumulation point)
• Upper and lower limits of a sequence

Theorems that you need to know how to prove

• The ε-characterization of the supremum and infimum (Theorem 1.15)
• Consequences of the field axioms (Theorems 1.3 -- 1.6)
• Consequences of the order axioms (Theorem 1.7)
• Every convergent sequence is bounded (Theorem 2.3)
• Arithmetic operations and limits (Theorem 2.4)
• Comparison theorems for limits (Theorem 2.5)
• Every bounded monotone sequence is convergent (Theorem 2.6)
• Principle of nested intervals (Theorem 2.7)
• Characterization of limits using subsequences (Theorems 2.10, 2.11)
• The Bolzano-Weierstrass principle and its corollaries (Theorems 2.12, 2.14, Corollary 2.14)
• Upper and lower limit are the greatest and the lowest subsequential limits (Theorems 1 and 2 discussed in class)
• Every Cauchy sequence of real numbers is convergent (Problem 13, 2.3)
• Every convergent sequence is Cauchy (Problem 14, 2.3)

General rules

• All tests are closed books/notes; graphing calculators, cell phones or laptops are not permitted. A basic scientific calculator is OK (example: TI 30X IIS or similar) but it is not required to answer the questions.
• The length of the test is 1 hr 15 minutes.
• Problems will come in the same format as on the quizzes: the expectation is that you write a clear and concise proof for the statement given in the question.