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Unformatted text preview: Math 240A: Real Analysis, Fall 2011 Additional Exercise Problems 1. Let ( X,ρ ) be a metric space. Define d : X × X → R by d ( x,y ) = ρ ( x,y ) 1 + ρ ( x,y ) ∀ x,y ∈ X. Prove that ( X,d ) is also a metric space. 2. Let ( X,ρ ) be a metric space and E a closed subset of X . Define d : X → R by d ( x ) = inf y ∈ E ρ ( x,y ) . For each integer n ≥ 1 define f n : X → [0 , ∞ ) by f n ( x ) = 1 1 + nd ( x ) ∀ x ∈ X. Prove that 0 ≤ f n ≤ 1 for each n , { f n } ∞ n =1 is decreasing, and lim n →∞ f n ( x ) = χ E ( x ) for all x ∈ X. 3. Let ( X,ρ ) be a metric space. If a Cauchy sequence { x n } in X has a subsequence that converges to some x ∈ X . Then { x n } itself converges to x. 4. Prove that any discrete metric space is complete. 5. Let X and Y be two nonempty sets and f : X → Y a mapping. For any T ⊆ Y define f 1 ( T ) = { x ∈ X : f ( x ) ∈ T } . Let M = { f 1 ( E ) : E ⊆ Y } . Show that M is a σalgebra on X....
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This note was uploaded on 12/11/2011 for the course MATH 240a taught by Professor Rothschild,l during the Fall '08 term at UCSD.
 Fall '08
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