1873_solutions

N 1 such that the inequality x n 2 holds whenever n n

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Unformatted text preview: y is in the set S and it is clear that xn Ý. 2. Suppose that S is a nonempty set of real numbers and that  is an upper bound of S. Prove that the following conditions are equivalent: a. We have   sup S. b. There exists a sequence x n in S such that x n  as n Ý. In Exercise 7 of the exercises on closure we saw that an upper bound  of set S is close to S if and only if  is equal to sup S. We also saw in a recent theorem that a number  is close to a set S if and only if there exists a sequence in S that converges to . The present exercise follows at once from these two facts. Of course, we can write this exercise out directly if we wish. 3. Given a sequence x n that is frequently in a set S of real numbers, and given a partial limit x of the sequence x n , is it necessarily true that x S? No, this statement need not be true. If we define x n  1 n for each n then, although x n is frequently in the set 1 it has the partial limit 1 that is not close to 1 . 4. Prove that a set U of real numbers is open if and only if every sequence that converges to a member of U must be eventually in U. We know from a recent theorem that a set H is closed if and only if no sequence that is frequently in H can have a limit that doesn’t belong to H. Therefore if U is a set of real numbers then the set R U is closed if and only if no sequence with a limit in U can fail to be eventually in U. Of course the exercise can also be done directly. 5. Given that S is a set of real numbers and that x is a real number, prove that the following conditions are equivalent: a. The number x is a limit point of the set S. b. There exists a sequence x n in the set S x such that x n x. Solution: The assertion in this exercise follows at once from the corresponding theorem about limits of sequences and closure of a set, and from the fact that x is a limit point of S if and only if xS x. 143 6. Prove that if x n is a sequence of real numbers then the set of all partial limits of x n is closed. Suppose that x n is a sequence of real numbers and write the set of partial limits of x n as H. In order to show that H is closed we shall show that R H is open. Suppose that x R H. In other words, suppose that x is a number that is not a partial limit of x n . Choose a number  0 such that the condition xn x ,x  holds for at most finitely many integers n. Given any number y x , x  , it follows from the fact that x , x  is a neighborhood of y and the fact that x n belongs to this neighborhood of y for at most finitely many integers n that y is not a partial limit of x n . In other words, x ,x  RH and we have shown, as promised, that the set R H is open. 7. Suppose that A and B are nonempty sets of real numbers and that for every number x y B we have x  y. Prove that the following conditions are equivalent: A and every number a. We have sup A  inf B. b. There exists a sequence x n in the set A and a sequence y n in the set B such that y n n Ý. xn 0 as Solution: From the information gi...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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