21504hw3sol - June 24, 2004 MATH 215 Homework 3 Solutions...

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Unformatted text preview: June 24, 2004 MATH 215 Homework 3 Solutions 1. Let X = R with the absolute value metric. Let K be the set consisting of 0 and the numbers 1 /n , for n = 1 , 2 , 3 ,... . Show that K is compact directly from the definition, i.e. without using Heine-Borel theorem. Proof. Let C = { G α : α ∈ A } be an arbitrary open cover of K . There is α ∈ A such that ∈ G α . Since G α is open, there is r > 0 such that B r (0) ⊂ G α . Since lim n →∞ 1 n = 0, we can find a natural number N such that 1 r < N . Then for all n ≥ N we have 1 n- 0 = 1 n ≤ 1 N < r ⇒ 1 n ∈ B r (0) ⊂ G α . Since the collection C is an open covering of K , for the elements 1 1 , 1 2 ,..., 1 N- 1 of K , there are sets G α 1 , G α 2 ,...,G α N- 1 in C such that 1 1 ∈ G α 1 , 1 2 ∈ G α 2 ,... 1 N- 1 ∈ G α N- 1 . Then K = { |{z} ∈ G α } ∪ { 1 1 |{z} ∈ G α 1 , 1 2 |{z} ∈ G α 2 ,..., 1 N- 1 | {z } ∈ G α N- 1 , 1 N , 1 N + 1 ,......
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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21504hw3sol - June 24, 2004 MATH 215 Homework 3 Solutions...

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