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Unformatted text preview: N A i is countable. Note that x ∈ A if and only if x ∈ A i for some i ∈ N . (List the members of the union in an inﬁnite array, and use it to deﬁne a surjection f : N → A . Then apply a result we proved in class. ) 4. Problem 4 Page 268. Justify your answer in no more than 10 lines. 5. Problem 8 Page 268. Recall that S bounded means that there is a number M > 0 such that  x  ≤ M for all x ∈ S . Justify your answer in no more than 10 lines. 6. Let S be a nonempty set of real numbers which has a lower bound. LetS denote {x : x ∈ S } . In no more than 10 lines, show thatS has an upper bound. By the Least Upper Bound Property,S has a least upper bound α . In no more than 20 lines, show thatα is the greatest lower bound of S ....
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This note was uploaded on 10/15/2011 for the course MATH 23b taught by Professor Bonglian during the Spring '09 term at Brandeis.
 Spring '09
 BongLian
 Math

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