R 8. (a) Show that if x²ⁿ < y²ⁿ and y > 0, then  (See Problem 6.)

(b) Show that if x²ⁿ < y²ⁿ and y < 0, then 

(c) Use Parts (a) and (b) to show that if x²ⁿ < y²ⁿ, then 

R 9. Prove the following by mathematical induction:

(a) If a₁, a₂, ..., a_n are positive, so is a₁ + a₂ + ... + a_n.

(b) If a₁, a₂, ..., a_n are positive, so is a₁a₂...a_n.

(c) If a₁ < b₁, a₂ < b₂, ..., a_n < b_n, then a₁ + a₂ + ... + a_n < b₁ + b₂ + ... + b_n.

(d) If 0 < a₁ < b₁, 0 < a₂ < b₂, ..., 0 < a_n < b_n, then a₁a₂...a_n < b₁b₂...b_n.

R 10. Show that:

(a) 

(b) 

(See Problem 3.)

11. Given that show that:

(a) 

(b) 

(c) 

12. Find all the integers n such that 2n² - 3 < 8n.

13. (a) Given that 0 < a < b, show that a² < ab < b².

(b) Given that 0 < a < b, show that 3a² < a² + ab + b²< 3b².

14. (a) Given that a < b, show that 

(b) Given that a < b, show that 

15. Given that 0 < x < y, show the following:

(a) 

(b) 
 
 

Monday, June 22, 1998