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 x
2
F
, a
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.)