1873_solutions

See that the equation cos i sin n cos n i sin n also

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Unformatted text preview: e fact that x is an interior point of U, choose   0 such that x , x   U. Since U V we therefore have x , x   V. Therefore x is an interior point of V. 4. Suppose that x is a real number and that U R. Prove that the following two conditions are equivalent: a. The set U is a neighborhood of the number x. b. It is possible to find two numbers a and b such that x a, b U. Solution: To prove that condition a implies condition b we assume that U is a neighborhood of the number x. Choose   0 such that By defining a  x x , x   U.  and b  x   we obtain two numbers a and b such that a  b and such that x a, b U. Now to prove that condition b implies condition a we assume that condition b holds. Choose numbers a and b such that a  b and such that x a, b U. Using the fact that the interval a, b is a neighborhood of x, choose   0 such that x , x   a, b . Since x , x   U we have shown that U is a neighborhood of x. 5. Suppose that x and y are two different real numbers. Prove that it is possible to find a neighborhood U of x and a neighborhood V of y such that U V  . Solution: We may assume, without loss of generality, that x  y. Choose a number c between x and y. The intervals Ý, c and c, Ý are, respectively, neighborhoods of x and y and the intersection of these two intervals is empty. 6. Given that S is a set of real numbers and that x is an upper bound of S, explain why S cannot be a neighborhood of x. Solution: If  is any positive number then, since all of the numbers between x and x   must lie in R S, the interval x , x   cannot be included in S. 7. Given that a set S of real numbers is nonempty and bounded above, explain why neither S nor R neighborhood of sup S. S can be a Solution: We see from Exercise 6 that S is not a neighborhood of sup S. Now we observe that, 96 whenever   0, since the number x  fails to be an upper bound of S, there must be members of S in the interval x , x . Therefore, whenever   0, the interval x , x   fails to be included in the set R S and so R S must also fail to be a neighborhood of x. 8. Suppose that A and B are sets of real numbers and that x is an interior point of the set A Þ B. Is it true that x must either be an interior point of A or an interior point of B? Hint: The answer is no. Give an example to show what can go wrong. 9. Suppose that A and B are sets of real numbers and that x is an interior point both of A and of B. Is it true that x must be an interior point of the set A B? Yes it is true. Suppose that x is an interior point of both of the sets A and B. Choose a number  1  0 such that x 1, x  1 A and choose a number  2  0 such that x 2, x  2 B. We now define  to be the smaller of the two numbers  1 and  2 and we observe that x , x   A B. 10. Suppose that x and y are real numbers and that U is a neighborhood of y. Prove that the set V defined by V  xu u U is a neighborhood of the number x  y. Solution: We need to find a number   0 such that the i...
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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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