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Study guide for the final exam
(MATH 6101-090, Fall 2018)

This guide will be continuously updated till the last day of classes.
Last update: Wednesday, December 5, 2018 (final version)
  1. Definitions and axioms to remember:
    Some fundamental notions you should know from lectures and/or prerequisites: equivalence relation, partial order, total order, well-ordered set, ring, field.
    Dedekind cuts (Definition 1.6.1), rational Dedekind cuts (Definition 1.6.4), real numbers (Definition 1.7.1), relations and operations on real numbers (Definitions 1.7.2, 1.7.3 and 1.7.5), upper bound and lower bound properties (Definition 1.7.7). Axiom for the Real Numbers (Axiom 2.2.4), intervals (Definition 2.3.6), absolute value (Definition 2.3.8), generalized infinite decimal fractions (Definition 2.8.4), base p representation of real numbers (Definition 2.8.7), eventually repeating base p representation (Definition 2.8.9), sequence (Definition 8.2.1), convergence of sequences (Definition 8.2.2), bounded sequence (Definition 8.2.5), divergence to infinity (Definition 8.2.14), (strictly) increasing and decreasing sequences (Definition 8.3.1), subsequence (Definition 8.3.5), Cauchy sequence (Definition 8.3.10), Cantor set (Example 8.4.9), series (Definition 9.2.1), convergence and divergence of partial sums (Definition 9.2.2), absolute and conditional convergence (Definition 9.4.1). Limits (Definition 3.2.1), bounded functions (Definition 3.2.5), operations on functions (Definition 3.2.9), continuity of a function (at a point) (Definition 3.3.1), uniform continuity (Definition 3.4.1).

  2. Statements you should be able to prove:
    Dedekind cuts have the greatest lower bound property (Theorem 1.7.8), on any totally ordered set the greatest lower bound property is equivalent to the least upper bound property (Theorem 1.7.9), generalized decimal fractions with integer part zero (Lemma 2.8.3 and parts (1) through (3) of Lemma 2.8.5 ), limit of a sequence is unique (Lemma 8.2.3), a convergent sequence is bounded (Lemma 8.2.6), multiplying a bounded sequence with a sequence that goes to zero yields a sequence that goes to zero (Lemma 8.2.7), an increasing sequence that is bounded from above is convergent, increasing unbounded sequence diverges to infinity (Theorem 8.3.3), Monotone Convergence Theorem (Corollary 8.3.4), Bolzano-Weierstrass Theorem (Theorem 8.3.9), convergent sequences are Cauchy sequences (Theorem 8.3.12), Cauchy sequences are bounded (Lemma 8.3.14), Cauchy sequences are convergent (Theorem 8.3.15), Cauchy Completeness Theorem (Corollary 8.3.16), ratio of Fibonacci numbers converges to the golden ratio (Example 8.4.10), convergence and divergence of the geometric series (Example 9.2.4), divergence of the harmonic series (Example 9.2.4), Divergence Test (Theorem 9.2.5), series of positive summands is convergent precisely when it is bounded (Lemma 9.3.1), Comparison Test (Theorem 9.3.2, only in the case when N=0), alternating series test (Theorem 9.3.8), absolute convergent series are convergent (Theorem 9.4.3). Uniformly continuous function on a bounded domain is bounded (Theorem 3.4.5), continuous function on a closed bounded interval is bounded (Corollary 3.4.8), Extreme Value Theorem (Theorem 3.5.1).

  3. Statements you should be able to state (without proof):
    We may define negative and inverse on Dedekind cuts (Lemma 1.6.8), comparing a Dedekind cut with a rational Dedekind cut (Lemma 1.7.4), real numbers form an ordered field (Theorem 1.7.6), rational numbers may be considered a subset of real numbers (Theorem 1.7.10), principles of mathematical induction (Theorems 2.5.1, 2.5.4, basically be able to prove any statement by induction), definition by recursion (Theorem 2.5.5). Basic properties of the absolute value (Lemma 2.3.9), Heine-Borel theorem (Theorem 2.6.14), unique representation of natural numbers in base p (Theorem 2.8.2), real numbers with an eventually repeating base p representation are exactly the rational numbers (Theorem 2.8.10, be able to rewrite rational number in base 10 in eventually repeating from and vice versa) operations on convergent sequences (Theorem 8.2.9), inequalities and limits (Theorem 8.2.11), Squeeze Theorem for sequences (Theorem 8.2.12), subsequence of a convergent sequence converges to the same limit (Lemma 8.3.7), every sequence has a monotone subsequence (Lemma 8.3.8), if a Cauchy sequence has a convergent subsequence than it is convergent (Lemma 8.3.13), Nested Interval Theorem (Theorem 8.4.7), facts about the Cantor set (Example 8.4.9), the limit of a series is unique (Lemma 9.2.3), operations on convergent series (Theorem 9.2.6), Limit Comparison Test (Theorem 9.3.4), Example of an absolute convergent and of a conditionally convergent series (Example 9.4.2), outline example of a conditionally convergent series whose limit may change after rearranging (Example 9.4.11), rearranging an absolute convergent series does not change the limit (Theorem 9.4.12), a series that is only conditionally convergent may be rearranged to have any limit (Theorem 8.4.15). Uniqueness of a limit (Lemma 3.2.2), sign-preserving property of limits (Lemma 3.2.4), continuous function on a closed bounded interval is uniformly continuous (Theorem 3.4.4), a function that has a limit at c is bounded on a set obtained by removing c from some neighborhood of it (Lemma 3.2.7), operations preserving the existence of limits (Theorem 3.2.10), operations preserving continuity (Theorem 3.3.5),

  4. What to expect
    The exam will be closed book. The above guide is meant to help with the mandatory part. For the optional part prepare as if it was another midterm. The mandatory part will be as long as the midterm, the optional part will have only about 5 questions. On the mandatory part, Some questions may ask you to state and prove a theorem from the list above, others may be like the exercises from your homework assignments. Even if a statement is listed "without proof" above, you must remember the proof of those parts of it that were on a homework assignment! There may be questions where you have to decide about an example whether it has certain properties. (E.g. "Is this function continuous?")