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Study guide for the final exam
(MATH 6101-090, Fall 2014)

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

  1. Definitions and axioms to remember:
    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), 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), power series (Definition 9.5.1)

  2. Statements you should be able to prove:
    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), 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 3.4.8), Extreme Value Theorem (Theorem 3.5.1), 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), 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), the set of real numbers is uncountable (Theorem 8.4.8), ratio of Fibonacci numbers converges to the golden ratio (Example 8.4.10), limit of a series is unique (Lemma 9.2.3), in a convergent series the terms must converge to zero (Theorem 9.2.5), monotone convergence for series (Lemma 9.3.1), Alternating Series Test (Theorem 9.3.8), absolute convergent series are convergent (Theorem 9.4.3).

  3. Statements you should be able to state (without proof):
    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), continuity in terms of limits (Lemma 3.3.2), sign-preserving property of continuous functions (Theorem 3.3.4), composing continuous functions preserves continuity (Theorem 3.3.8), uniform continuity implies continuity (Lemma 3.4.2), Intermediate Value Theorem (Theorem 3.5.2), operations on sequences that preserve convergence (Theorem 8.2.9, be able to prove part (4)), inequalities and limits (Theorem 8.2.11), Squeeze Theorem for sequences (Theorem 8.2.13), every sequence contains a monotone subsequence (Lemma 8.3.8), Bolzano-Weierstrass Theorem (Theorem 8.3.9), Cauchy Completeness Theorem (Theorem 8.3.16), Sequential Characterization of Limits (Theorem 8.4.1), Sequential Characterization of Continuity (Corollary 8.4.2), Nested Interval Theorem (Theorem 8.4.7), operations preserving convergence (Theorem 9.2.6), Comparison Test (Theorem 9.3.2), Limit Comparison Test (Theorem 9.3.4), Ratio Test (Theorem 9.4.4), any rearrangement of an absolutely convergent series in absolutely convergent and has the same limit (Theorem 9.4.12), power series have a convergence radius (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?")