1873_solutions

Intersection of all of the sets h n must also be

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Unformatted text preview: ercies that appear here require the student to be familiar with the concept of 1. a. Given that S is an uncountable set of real numbers, explain why there must exist an integer n such that the set S n, n  1 is uncountable. Solution: If the set S n, n  1 were countable for every n then, since S Ý S n, n  1 , n Ý the set S, being the union of a countable family of countable sets, would be countable by an earlier theorem. b. Prove that every uncountable set of real numbers must have a limit point. Hint: Apply part a and the Bolzano-Weierstrass theorem to an uncountable set of the form S 2. n, n  1 . a. Suppose that S is a set of real numbers, that  0 and that for any two different members x and t of the set S we have |x t | . Prove that S is a countable set. b. Suppose that S is an uncountable set of real numbers and, for each positive integer n, suppose that 1 for every y S x. S n  x S |x y | n Prove that each set S n is countable and that LS  S Ý  Sn. n1 c. Improve Exercise 1b by showing that if S is an uncountable set of real numbers then S has an uncountable set of limit points that belong to S. Solution: We deduce from part b that the set S S S and since the set S is countable, the set S 3. Suppose that S is a set of real numbers and that LS L S is countable. Therefore, since ÞS LS L S must be uncountable.  0. For each integer n, in the event that the set 152 xSn x n1 is nonempty, suppose that a number x n has been chosen in this set. Prove that for every member x of the set S there is an integer n for which the number x n is defined and for which x xn , xn  .  0 then there exists a countable subset E of S such that 4. Prove that if S is any set of real numbers and  xn S , xn  . xE 5. Suppose that S is a set of real numbers. a. Prove that for every positive integer n it is possible to find a countable subset E n of the set S such that S  x 1 ,x  1 n n x En b. Prove that there exists a countable subset E of S such that S E. 6. Prove that every closed subset of R is the closure of some countable set. Exercises on Upper and Lower Limits 1. Prove that if x n is a sequence of real numbers then x n has a limit if and only if lmsup x n  lminf x n . nÝ nÝ Solution: This exercises follows at once from an earlier theorem. 2. Prove that a sequence x n of real numbers is bounded above if and only if lmsup x n  Ý. nÝ We know that a sequence is bounded above if and only if Ý is not a partial limit of the sequence. There a sequence is bounded above if and only if its largest partial limit is not Ý. 3. Suppose that x n is a sequence of real numbers. a. Prove that if x is a partial limit of the sequence x n then the number x is a partial limit of the sequence xn . if Ý is a partial limit of a sequence x n then x n is unbounded above and so x n is unbounded below, making Ý a partial limit of x n . Now suppose that a real number x is a partial limit of a given sequence x n . To show that the number x is a partial limit of the sequence x...
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