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Study guide for the final exam
(MATH 6101-090, Fall 2016)

This guide will be continuously updated till the last day of classes.

  1. Definitions and axioms to remember:
    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), 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), one-sided limits (Definition 3.2.15), continuity of a function (at a point) (Definition 3.3.1), uniform continuity (Definition 3.4.1), power series (Definition 9.5.1).

  2. Statements you should be able to prove:
    Archimedean property (Theorem 2.6.7), Heine-Borel theorem (Theorem 2.6.14), 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), product of convergent sequences is convergent (part (4) of Theorem 8.2.9), inequalities and limits (Theorem 8.2.11), Squeeze Theorem for sequences (Theorem 8.2.12), 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), subsequence of a convergent sequence converges to the same limit (Lemma 8.3.7), 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 Completeness Theorem (Corollary 8.3.16), ratio of Fibonacci numbers converges to the golden ratio (Example 8.4.10), Divergence Test (Theorem 9.2.5), series of positive summands is convergent precisely when it is bounded (Lemma 9.3.1), Comparison Test (Theorem 9.3.1), alternating series test (Theorem 9.3.8), absolute convergent series are convergent (Theorem 9.4.3), Uniqueness of a limit (Lemma 3.2.2), 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), continuity in terms of limits (Lemma 3.3.2), sign-preserving property of continuous functions (Theorem 3.3.4), operations preserving continuity (Theorem 3.3.5), continuous function on a closed bounded interval is uniformly continuous (Theorem 3.4.4), uniformly continuous function on a bounded domain is bounded (Theorem 3.4.5), continuous function on a closed bounded interval is bounded (Corollary 3.4.6), Extreme Value Theorem (Theorem 3.5.1), Sequential Characterization of Limits (Theorem 8.4.1).

  3. Statements you should be able to state (without proof):
    Basic properties of the absolute value (Lemma 2.3.9), 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), 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), operations on convergent series (Theorem 9.2.6), Limit Comparison Test (Theorem 9.3.4), Ratio Test (Theorem 9.4.4), 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), Sign-preserving property of limits (Theorem 3.2.4), the product of a bounded function and of a function that goes to zero also goes to zero (Lemma 3.2.8), composition and limits (Theorem 3.2.12), inequalities and limits (Theorem 3.2.13), Squeeze Theorem (Theorem 3.2.14), limits in terms of one-sided limits (Lemma 3.2.17), composing continuous functions preserves continuity (Theorem 3.3.8), Pasting Lemma (Lemma 3.3.10), uniform continuity implies continuity (Lemma 3.4.2), Intermediate Value Theorem (Theorem 3.5.2), Sequential Characterization of Continuity (Corollary 8.4.2), results on the radius of convergence (Lemma 9.5.3 and Theorem 9.5.4).

  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?")