homework6solnmat25

# homework6solnmat25 - Homework 6 Solutions 1 Consider the...

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Unformatted text preview: Homework 6 Solutions 1. Consider the set S = [0 , 3) ∪ (3 , 5). What is the interior of S ? What are the limit points of S ? What is the closure of S ? Solution: The interior of S is the set (0 , 3) ∪ (3 , 5), the set of limit points is [0 , 5], so the closure of S is [0 , 5]. 2. If A is open and B is closed, prove that A \ B is open. Solution: Let x ∈ A \ B . We want to show that ∃ > 0 so that B ( x ) ⊆ A \ B . Since A is open, there exists an 1 so that B 1 ( x ) ⊆ A . Now B is closed, so B c is open. Therefore, since x ∈ B c , there exists an 2 so that B 2 ( x ) ⊆ B c . Then let = min { 1 , 2 } . Then B ( x ) = B 1 ( x ) ∩ B 2 ( x ) ⊆ A ∩ B c = A \ B. 3. Let S be a bounded infinite set and let x = sup S . Prove that either x ∈ S or x is a limit point of S . Solution: Suppose x 6∈ S . Because x = sup S , ∀ > 0, x- is not an upper bound of S , so there is some element s of S so that x- < s < x . But that exactly says that ∀ > 0, B ( x ) ∩ S 6 = ∅ , which is the definition of a limit point., which is the definition of a limit point....
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homework6solnmat25 - Homework 6 Solutions 1 Consider the...

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