1873_solutions

Bounded sets of real numbers that for every x a there

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: 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  qx 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...
View Full Document

Ask a homework question - tutors are online