EXERCISE 4. 18. Let n' b Z with 0-b n, and let EEA(n, b)-(ri, Si, ti) This exercise develops some key properties of the fractions -si/ti as approxima- tions to b. For i = 1, . . . , + 1, let := b + si/ti (a) Show that = ri/tin for i = 1, . . . , +1. (b) Show that successive 's strictly decrease in absolute value, and alternate in sign (c) Show that ! 1m for i = I, . . . , , and +1 0 (d) Show that for all s't E Z with t 0, if lb-s/t| 1/2t2, then s/t =-si/ti for some i = 1, . . . , + 1 . Hint: use part (ii) of Theorem 4.9. (e) Consider a fixed index i e 12, , + 1 } . Show that for all s,t Z, if 011 Itil and b-s/t| |eil, then s/t =-s/ti. In this sense,-s/ti is the unique, best approximation to b among all fractions of denominator at most [til. Hint: use part (i) of Theorem 4.9. EXERCISE 4. 18. Let n' b Z with 0-b n, and let EEA(n, b)-(ri, Si, ti) This exercise develops some key properties of the fractions -si/ti as approxima- tions to b. For i = 1, . . . , + 1, let := b + si/ti (a) Show that = ri/tin for i = 1, . . . , +1. (b) Show that successive 's strictly decrease in absolute value, and alternate in sign (c) Show that ! 1m for i = I, . . . , , and +1 0 (d) Show that for all s't E Z with t 0, if lb-s/t| 1/2t2, then s/t =-si/ti for some i = 1, . . . , + 1 . Hint: use part (ii) of Theorem 4.9. (e) Consider a fixed index i e 12, , + 1 } . Show that for all s,t Z, if 011 Itil and b-s/t| |eil, then s/t =-s/ti. In this sense,-s/ti is the unique, best approximation to b among all fractions of denominator at most [til. Hint: use part (i) of Theorem 4.9