1873_solutions

# Bounded sets of real numbers that for every x a there

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Unformatted text preview: it follows from the facts that x x sup A  inf B  y  which tells us that x   y. Therefore it is impossible to find x A and y have reached the desired contradiction. sup A and inf B y B such that x    y and we 8. Suppose that A is a nonempty bounded set of real numbers, that A has no largest member and that x  sup A. Prove that there are at least two different members of A lying between x and sup A. Since x is less than the least upper bound of A we know that x can’t be an upper bound of A. Using the fact that x isn’t an upper bound of A we choose a member a of A such that x  a. Since A has no largest member it must have a member larger than the member a. We choose a member b of A such that a  b. Since A has members greater than b we deduce that x  a  b  sup A and we have found two different members of A between x and sup A. 9. Suppose that A is a nonempty bounded set of real numbers, that   0 and that for any two different members x and y of A we have |x y | . Prove that A has a largest member. You can find a hint to the solution of this exercise in a forthcoming theorem. 10. Suppose that S is a nonempty bounded set of real numbers, that   inf S and   sup S, and that every number that lies between two members of S must also belong to S. Prove that S must be one of the four intervals ,  , ,  , ,  , ,  . See a coming theorem for a solution of this exercise. 11. Suppose that A is a nonempty bounded set of real numbers, that   inf A and that   sup A. Suppose that S  x y x A and y A . Prove that sup S   . You will find a solution to this exercise in the next section. 12. Suppose that A is a set of numbers and that A is nonempty and bounded above. Suppose that q is a given number and that the set C is defined as follows: C  qx x A . Prove that the set C is nonempty and bounded above and that sup C  q  sup A This exercise is a special case of the next exercise because C  A  q . 13. Suppose that A and B are nonempty bounded sets of numbers and that the sets A  B and A above. Prove that sup A  B  sup A  sup B and sup A B  sup A inf B B are defined as Solution: We shall prove that sup A  B  sup A  sup B Step 1: We want to show that the number sup A  sup B is an upper bound of the set A  B. Suppose that 81 x A  B. Using the definition of A  B we choose a number a A and a number b x  a  b. Since a sup A and b sup B we see that x  a  b sup A  sup B. B such that Step 2: We want to show that the number sup A  sup B is actually the least upper bound of the set A  B. Suppose that u is any upper bound of the set A  B. Given any number a A and b B we have a  b u. Therefore, whenever b B we know that the inequality aub holds for all a A. Therefore, whenever b B, the number u b is an upper bound of A and must satisfy the condition sup A u b which we can also write as b u sup A. We conclude that u sup A is an upper bound of B and so sup B u sup A which we can write as sup B  sup A u. Thus sup A  sup...
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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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