## Math 355: Introduction to Analysis Study Guide for Exam 1

The first midterm exam is on Friday, March 2, during the usual class period.

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

Please be sure to review homework and example problems for the chapters given below!

1.1-1.2: Know the notation, theorem and proof techniques from these preliminary sections.

1.3-1.4: You should know what we mean by upper bound, supremum, lower bound, infimum and the Axiom of Completeness. Make sure you know the statements (and proofs) of Theorem 1.3.8 (the alternative definition of supremum), Theorem 1.4.1 (the Nested Interval Property), Theorem 1.4.2 (Archimedean Property), Theorem 1.4.3 (density of rationals) and Corollary 1.4.4 (density of irrationals).

1.5-1.6: Know what we mean by a 1-1 correspondence or bijection (and how to show a function is a bijection either by explicitly showing it is 1-1 and onto, or by finding an explicit inverse function). Know the definition of cardinality and countability. Be familiar with the class and textbook examples and know how to show whether a given set is countable or uncountable. In particular, you should know how to show that the rationals are countable and the reals are uncountable.

2.2-2.3: You should know what a sequence is and what it means for it to converge or diverge. Know how to show that a sequence converges directly using the definition, and also by the Algebraic Limit Theorem (Theorem 2.3.3) and the Squeeze Theorem (Exercise 2.3.3). Know what it means for a sequence to be bounded, and the statement and proofs of Theorems 2.3.2 and 2.3.4.

2.4: Know the statement and proof of the Monotone Convergence Theorem, and how to apply it, especially in problems such as Exercise 2.4.1. You should also know what an infinite series is, the associated sequences of terms and partials sums, and what it means for a series to converge or diverge. You should know the statement of Corollary of 2.4.7, but you do not need to know the Cauchy Condensation Test (Theorem 2.4.6) for this exam.

2.5: Know the definition of a subsequence and the divergence criterion given by Theorem 2.5.2. (See also, Example 2.5.4.) Know the statement and proof of the Bolzano-Weierstrass Theorem.

2.6: Know what we mean by a Cauchy sequence, and the statement and proof of Theorem 2.6.4 (the Cauchy Criterion, which says that a sequence converges iff it is a Cauchy sequence). The diagram at the top of p. 69 is a convenient way to remember how we proved several important theorems, but you do not need to know the rest of the discussion in the "Completeness Revisited" subsection.

2.7: Know what it means for a series to converge absolutely or conditionally. Make sure you know the following convergence tests for series given in this section: Theorem 2.7.1 (Algebraic Limit Theorem), Theorem 2.7.2 (Cauchy Criterion), Theorem 2.7.3 (Divergence Criterion), Theorem 2.7.4 (Comparison Test), Example 2.7.5 (Geometric Series), Theorem 2.7.6 (Absolute Convergence Test), Theorem 2.7.7 (Alternating Series Test) and Exercise 2.7.9 (Ratio Test). You do not need to know Theorem 2.7.10 for this exam.

3.2: Know the definition of the terms ε-neighbourhood, limit point, isolated point, open set, closed set, closure, and complement. Know the statements and proofs of Theorems 3.2.3, 3.2.5, 3.2.8, 3.2.10, 3.2.12, 3.2.13 and 3.2.14.