Econ511-WTK-2008Mid - 4111 R N Notes • Theorem 4 R N is...

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Econ 511 Professor: John Nachbar Fall 2008 Required Theorems for 511 Midterm The following is a list of theorems that may appear as test questions. You should be able to prove any one of these. Note that some results are from the 4111 notes and some from the 511 notes. This is not an exhaustive list of important results. 4111 Metric Space Notes Theorem 5 (Convergent sequences are Cauchy) Theorem 6 (Cauchy sequences with a convergent subsequence are convergent) Theorem 7 (A set is open iff every point is interior) Theorem 11 (3 equivalent characterizations of closedness) 4111 Compactness Notes Theorem 1 (Sequentially compact equivalent to complete and totally bounded)
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Unformatted text preview: 4111 R N Notes • Theorem 4 ( R N is complete, using previously proved theorems) • Theorem 6 (Heine-Borel, again using previously proved theorems) 4111 R ∞ Notes • Theorem 6 (Examples that bounded sets in R ∞ need not be totally bounded) 511 Continuity Notes • Theorem 4 (Equivalence of the ε-δ and sequence definitions of continuity) 511 Continuity and Connectedness Notes • Theorem 2 (Intermediate Value Theorem) 4111 Existence of Optima Notes • Theorem 1(the continuous image of a compact set is compact) • Theorem 2 (a continuous real-valued function on a compact set has a maxi-mum)...
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