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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.)