HW 7
Due Friday, June 22, 2007
Note: Only six of these problems will be graded for one point each.
1. Let a sequence { x n
} of numbers be defined recursively by x
1
= 0 and x n
+1
= x n
2
+ 1
.
Prove by induction that x n
≤ x n
+1 for all n ∈ N . Would this conclusion change if we set x
1
= 2?
2. For each n ∈ N , let P n denote the assertion ” n 2 + 5 n + 1 is an even integer.”
(a) Prove that P n
+1 is true whenever P n is true.
(b) For which n is P n actually true? What is the moral of this exercise?
3. Use Theorem 24 to prove that n 2 < 2 n for all n ≥ 5.
4. Let x, y ∈ R . Prove xy = 0 iff x = 0 or y = 0.
5. Let x, y, z ∈ R . Prove that if x < y and z < 0, then xz > yz .
6. Let x, y ∈ R . Prove | x · y | = | x | · | y | .
7. (a) Prove that | a + b + c | ≤ | a | + | b | + | c | for all a, b, c ∈ R .
(b) Use induction to prove | a
1
+ a
2
+ a
3
+ · · · + a n
| ≤ | a
1
| + | a
2
| + · · · + | a n
| for n numbers a
1
, a
2
, . . . , a n
.
8. Prove that in any ordered field F , a 2 x 2
+ 1 > 0 for all a ∈ F . Conclude from this that if the equation
+ 1 = 0 has a solution in a field, then that field cannot be ordered. (Thus, it is not possible to define an order relation on the set of all complex numbers that will make it an ordered field.)