1873_solutions

W 2 x 2 y 2 since the numbers x 1 x 1 and x 1 x 2

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: any number x, the interval x 1, x  1 can contain at most two integers. We know that a neighborhood of a limit point of a set must always contain infinitely many members of that set and so we conclude that no number x can be a limit point of the set Z of integers. 2. Prove that L Q  R. Suppose that x is any real number. To show that x is a limit point of Q, suppose that   0. Since there are rational numbers in the interval x, x   we know that the set x , x   Q x . 3. Prove that L 1 n Z  0 . n If x is any negative number then the interval Ý, 0 is a neighborhood of x that fails to contain any members of the set 1 n Z  . Thus a negative number can not be a limit point of n 1  nZ. n x If x is any positive number then the interval 2 , Ý is a neighborhood of x which fails to contain 1 infinitely many members of the set n n Z  . To see why, note that if n is a positive integer then the condition 1 x ,Ý n 2 can hold only if n  2 . Therefore no positive number can be a limit point of 1 n Z . x n 1  Finally, we need to explain why 0 must be a limit point of n n Z . Suppose that   0. Choose an integer k  1 and observe that  1 1 0 , 0   n Z 0 n k 1 0 must be nonempty. n Z from which we deduce that the set 0 , 0   n 4. a. Give an example of an infinite set that has no limit point. As we saw in Exercise 1, the infinite set Z has no limit point. b. Give an example of a bounded set that has no limit point. A finite set like 2 will not have any limit points. We could also look at the empty set . c. Give an example of an unbounded set that has no limit point. 104 As we saw in Exercise 1, the infinite set Z has no limit point. d. Give an example of an unbounded set that has exactly one limit point. n Z  has only the limit point 0. The unbounded set Z Þ 1 n e. Give an example of an unbounded set that has exactly two limit points. The set ZÞ 1 n Z Þ 1  1 n Z n n has the two limit points 0 and 1. We can see this directly or we can use the assertion proved in Exercise 6 below. 5. Prove that if A and B are sets of real numbers and if A B then L A LB. Suppose that A and B are sets of real numbers and that A B. Suppose that x is a limit point of A. We need to explain why x has to be a limit point of B. Suppose that   0. Since the set x , x   A x is nonempty and since x , x   A x x , x   B x we deduce that the set x , x   B x is nonempty. 6. Prove that if A and B are sets of real numbers then L AÞB  L A ÞL B . Solution: Since A LB L A Þ B . Thus A Þ B we know that L A L A Þ B and similarly we know that L A ÞL B L AÞB . Now suppose that a number x fails to belong to the set L A Þ L B . Choose a number  1  0 such that the interval x  1 , x   1 contains only finitely many members of the set A. Choose a number  2  0 such that the interval x  2 , x   2 contains only finitely many members of the set B. We now define  to be the smaller of the two numbers  1 and  2 and we observe that, although   0, the inte...
View Full Document

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.

Ask a homework question - tutors are online