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Unformatted text preview: 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 - 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 - 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 - 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 -neighborhood on which f ( x ) > 0): useful theorem and illustrative proof. 4.3 Not necessary to review for test. Check exercises....
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