Hints for Selected Exercises Hints For Selected Exercises of Chapter A Exercise 7. Let me show that if R is transitive, then xPR y and yRz implies xPR z. Since PR ⊆ R, it is plain that xRz holds in this case. Moreover, if zRx holds as well, then yRx, for R is transitive and yRz. | Hints for Selected Exercises Hints For Selected Exercises of Chapter A Exercise 7. Let me show that if R is transitive then xP y and yRz implies xPrZ. Since Pr C R it is plain that xRz holds in this case. Moreover if zRx holds as well then yRx for R is transitive and yRz. But this contradicts xP y. Exercise 13. c Suppose c S 0 which is possible only if S 3 . Take any x1 E S. Since c . S 0 there is an x2 E S xi with x2 - x1. Similarly there is an x3 E S x1 x with x y x . Continuing this way I find S x1 . x s with x s S- S- x1. Now find a contradiction to being acyclic. Exercise 14. Apply Sziplrajn s Theorem to the transitive closure of the relation p P U xj x Y . Exercise 16. e inf A P A and sup A P B E X J A C B for any class A C X. Exercise 18. Define the equivalence relation on X by x y iff f x f y let Z X and let g be the associated quotient map. Exercise 20. If f was such a surjection we would have f x y E X y f y for some x E X. Exercise 29. Show that inf S sup - s E R s E S . Exercise 30. Consider first the case where a 0. Apply Proposition 6. b twice to find a b E Q such that 0 a a b b. Now x E a b iff 0 X a 1 while b -a 75 E 0 1 Q. Why Exercise 36. ym a zm xm xm a for each m. Exercise 38. For every m E N there exists an xm E S with xm X sup S. Exercise 40. b By part a and the Bolzano-Weierstrass Theorem every real Cauchy sequence has a convergent subsequence. Exercise 41. Note that yk yk 1 yk xkl1 xkl xk l xk l yk for all k k l E N. Use i and ii and a 3x argument to establish that yk is Cauchy. So by the previous exercise yk x for some x E R. Now use the inequality xkl x xkl yk yk x k l 1 2 . to conclude that xkl x. ill 11 1 1 Exercise 43. How about 1 1 2 1 1 2 .j and 1 1 2 1 1 2 .j 563 Exercise 45. a This follows from the fact that x i x 2 - x xi for all k E N. By the way the converse of a is true too just verify that 52 xi is a real Cauchy sequence Exercise 40 . Exercise 48. a Try the sequence -1 m . b Suppose that 52TO Xj x E R. Then sm x so for any to y
