118Homework5 - I k ’s by nonempty open bounded intervals...

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Homework 5 Due Monday, Sept. 27 1. A closed, bounded interval I is a nonempty subset of R of the form I := [ a, b ] = { x R : a x b } . We say that the length of I is b - a . Prove that if I 1 I 2 I 3 . . . is a nested sequence of closed, bounded intervals of R with the length of I k approaching 0 as k → ∞ , then T k =1 I k consists of a single point. 2. Provide an example that shows that the intersection in Problem 1 may be empty if we replace the
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Unformatted text preview: I k ’s by nonempty open, bounded intervals O k := ( a, b ) = { x ∈ R : a < x < b } . 3. A rectangle R is a subset of R 2 of the form R = I × J = { ( x, y ) : x ∈ I and y ∈ J } for some closed, bounded intervals I, J ⊆ R . Prove that if R 1 ⊇ R 2 ⊇ R 3 ⊇ . . . is a nested sequence of rectangles, then T ∞ k =1 R k is nonempty....
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