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# review2 - MAT 320 Fall 2007 Theorem be able to apply Review...

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MAT 320 Fall 2007 Review for Midterm 2 Theorem: be able to apply Theorem : and know what goes into the proof Theorem : and be able to prove. § 3.3 Monotone Convergence Theorem . The least upper bound property is crucial. Understand Example 3.3.3(b). § 3.4 Theorem 3.4.4. Monotone Subsequence Theorem, Bolzano- Weier- strass Theorem (first proof) . § 3.5 Know definition of Cauchy sequence. Cauchy Convergence Crite- rion . Proof uses Lemma : a Cauchy sequence is bounded; then Bolzano- Weierstrass to produce a candidate limit; then additional - δ argument to show that limit works. Know definition of contractive sequence. A contractive sequence is a Cauchy sequence . Proof is straightforward once you use a k + a k +1 + · · · + a k + = a k (1 - a +1 ) / (1 - a ). § 3.6 Know definition of properly divergent sequence. § 4.1 Know definition of cluster point and of limit of a function at a cluster point. Theorem 4.1.5 (uniqueness of limit): basic and paradigmatic - δ argument. Sequential Criterion for Limits (Theorem 4.1.8) . Divergence Criteria (4.1.9). § 4.2 Theorem 4.2.2 ( f has a limit at c implies f is bounded on a neigh- borhood of c ). Nice - δ argument. Theorem 4.2.3 on limits of sums, products, quotients. Theorem 4.2.6 on -inequalities persisting to limits. Theorem 4.2.9 (lim x c f ( x ) > 0 implies that c has a δ -neighborhood on which f ( x ) > 0): useful theorem and illustrative proof. § 4.3 Not necessary to review for test. Check exercises. § 5.1 Know definition of “ f continuous at c ” in - δ terms. Understand Re- mark after Theorem 5.1.2. Sequential Criterion for Continuity . Know 5.1.6 Examples (g) and (h).

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