511-Midterm-2008F

# 511-Midterm-2008F - 4 Theorem Let X,d x and Y,d y be metric...

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Economics 511 Professor: John Nachbar Fall 2008 511 Midterm You have until 11:30AM. You can use either pen or pencil but write legibly and with good syntax. A “correct” but unintelligible answer is a wrong answer. Prove the following. 1. Theorem Let ( X, d ) be a metric space. Let { x t } be a Cauchy sequence in X . If { x t } has a convergent subsequence then it is convergent. 2. Theorem. Let ( X, d x ) be a metric space. If a set C X is complete and totally bounded then it is sequentially compact. 3. Theorem. Let C R N . C is compact iff it is closed and bounded.
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Unformatted text preview: 4. Theorem. Let ( X,d x ) and ( Y,d y ) be metric spaces. For any non-empty set C ⊆ X , if f : C → Y is continuous and C is compact then f ( C ) is compact. 5. Theorem. Let ( X,d x ) and ( Y,d y ) be metric spaces and suppose f : X → Y and g : X → Y are both continuous. Let E ⊆ X be dense in X . (a) If { y t } is a sequence in Y and y t converges both to y * and to ˆ y then y * = ˆ y . (b) If f = g on E then f = g on X . (c) f ( E ) is dense in f ( X ) ....
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