xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

数学代写-3283W

时间：2020-12-16

Math 3283W

Fall 2020

Time Limit: 120 minutes

This exam contains 8 numbered problems. Point values are in parentheses. No books, notes, or

electronic devices are allowed.

As on the writing quizzes, your work will be graded on the quality of your writing as well as on the

validity of the mathematics, but on this final exam there will be no separate writing score for any

of the problems. It is already included in the total points.

Do not use symbols for logical connectives and quantifiers unless directed to do so. That is, avoid

the use of the symbols ⇒, ⇔, ∧, ∨, ∼, ∀, ∃, and 3.

Axioms for Ordered Fields

A1. For all x, y ∈ F, x+ y ∈ R, and if x = w and y = z then

x + y = w + z.

A2. For all x, y ∈ F, x + y = y + x.

A3. For all x, y, z ∈ F, x + (y + z) = (x + y) + z.

A4. There is a unique element 0 ∈ F such that x+ 0 = x for

all x ∈ F.

A5. For each x ∈ F, there exists a unique element y such

that x + y = 0. (Often, we write y = −x.)

M1. For all x, y ∈ F, x · y ∈ R, and if x = w and y = z then

x · y = w · z.

M2. For all x, y ∈ F, x · y = y · x.

M3. For all x, y, z ∈ F, x · (y · z) = (x · y) · z.

M4. There is a unique element 1 6= 0 in F such that x ·1 = x

for all x ∈ F.

M5. For each x 6= 0 in F, there exists a unique element y

such that x · y = 1. (Often, we write y = 1/x.)

DL. For all x, y, z ∈ F, x · (y + z) = x · y + x · z.

O1. There is a relation < such that, for all x, y ∈ F, exactly

one relation holds: x = y, x < y or y < x.

O2. For all x, y, z ∈ F, if x < y and y < z then x < z.

O3. For all x, y, z ∈ F, if x < y then x + z < y + z.

O4. For all x, y, z ∈ F, if x < y and 0 < z then x · z < y · z.

1. (12 points) Answer “Yes” or “No” and justify your answer to the following questions. You

may quote theorems from the book in your responses.

(a) Does there exist an surjective function from A = {1, 2} to B = {3, 4, 5}?

(b) Is the set of all irrationals in R countable?

(c) Does the set S =

{

n2 − 2n

3n− 7

∣∣∣n ≥ 2, n ∈ N} have a finite supremum?

2. (16 points) For each of the following subsets of R, list the set of boundary points, accumulation

points, and whether S is open, closed, or neither. (No justification is required in this question.)

(a)

⋃∞

n=1(0, n)

(b) (1, 4] ∩ (2, 6]

(c) N

(d) {(−1)n n+1n }

3. (8 points) Use the axioms in an ordered field (see cover page) to show that in any ordered field

F , given elements x and y in F , if x < y, then −y < −x.

4. (10 points) Prove that, for all natural numbers n,

1

1 · 2 +

1

2 · 3 +

1

3 · 4 + · · ·+

1

n(n + 1)

=

n

n + 1

5. (12 points) Find limx→1 f(x) for the function defined by

f(x) =

{

2x2−3x+1

x−1 for x 6= 1

5 for x = 1.

and prove your answer is correct using the definition of the limit of a function. (Do not rely on

any other results about limits from class.)

6. (12 points) Let f : A → B be any function and let D1, D2 be any non-empty subsets of B.

Proof or counterexample:

f−1(D1 ∪D2) = f−1(D1) ∪ f−1(D2)

7. (14 points) (a) State the definition of compactness (using the notion of open covers)

(b) State the Heine-Borel theorem

(c) Complete the following proof of the Bolzano-Weierstrass theorem (which states that every

bounded, infinite subset of R has an accumulation point):

Page 2

Proof by contradiction: Let S be a bounded set with infinitely many points, and such that S

has no accumulation points. Thus, S is closed, since it trivially contains all of its accumulation

points. So by the Heine-Borel theorem, S is compact.

Since S has no accumulation points, every point x ∈ S has a neighborhood Nx with Nx ∩ S =

{x}. Thus the set {Nx, x ∈ S} are an open cover. But...

8. (12 points) Find the limit of the sequence {sn} defined according to the relations s1 = 1 and

sn+1 =

1

5(sn + 7) for all n ≥ 1. Prove your answer is correct using the Monotone Convergence

Theorem.

Page 3

Fall 2020

Time Limit: 120 minutes

This exam contains 8 numbered problems. Point values are in parentheses. No books, notes, or

electronic devices are allowed.

As on the writing quizzes, your work will be graded on the quality of your writing as well as on the

validity of the mathematics, but on this final exam there will be no separate writing score for any

of the problems. It is already included in the total points.

Do not use symbols for logical connectives and quantifiers unless directed to do so. That is, avoid

the use of the symbols ⇒, ⇔, ∧, ∨, ∼, ∀, ∃, and 3.

Axioms for Ordered Fields

A1. For all x, y ∈ F, x+ y ∈ R, and if x = w and y = z then

x + y = w + z.

A2. For all x, y ∈ F, x + y = y + x.

A3. For all x, y, z ∈ F, x + (y + z) = (x + y) + z.

A4. There is a unique element 0 ∈ F such that x+ 0 = x for

all x ∈ F.

A5. For each x ∈ F, there exists a unique element y such

that x + y = 0. (Often, we write y = −x.)

M1. For all x, y ∈ F, x · y ∈ R, and if x = w and y = z then

x · y = w · z.

M2. For all x, y ∈ F, x · y = y · x.

M3. For all x, y, z ∈ F, x · (y · z) = (x · y) · z.

M4. There is a unique element 1 6= 0 in F such that x ·1 = x

for all x ∈ F.

M5. For each x 6= 0 in F, there exists a unique element y

such that x · y = 1. (Often, we write y = 1/x.)

DL. For all x, y, z ∈ F, x · (y + z) = x · y + x · z.

O1. There is a relation < such that, for all x, y ∈ F, exactly

one relation holds: x = y, x < y or y < x.

O2. For all x, y, z ∈ F, if x < y and y < z then x < z.

O3. For all x, y, z ∈ F, if x < y then x + z < y + z.

O4. For all x, y, z ∈ F, if x < y and 0 < z then x · z < y · z.

1. (12 points) Answer “Yes” or “No” and justify your answer to the following questions. You

may quote theorems from the book in your responses.

(a) Does there exist an surjective function from A = {1, 2} to B = {3, 4, 5}?

(b) Is the set of all irrationals in R countable?

(c) Does the set S =

{

n2 − 2n

3n− 7

∣∣∣n ≥ 2, n ∈ N} have a finite supremum?

2. (16 points) For each of the following subsets of R, list the set of boundary points, accumulation

points, and whether S is open, closed, or neither. (No justification is required in this question.)

(a)

⋃∞

n=1(0, n)

(b) (1, 4] ∩ (2, 6]

(c) N

(d) {(−1)n n+1n }

3. (8 points) Use the axioms in an ordered field (see cover page) to show that in any ordered field

F , given elements x and y in F , if x < y, then −y < −x.

4. (10 points) Prove that, for all natural numbers n,

1

1 · 2 +

1

2 · 3 +

1

3 · 4 + · · ·+

1

n(n + 1)

=

n

n + 1

5. (12 points) Find limx→1 f(x) for the function defined by

f(x) =

{

2x2−3x+1

x−1 for x 6= 1

5 for x = 1.

and prove your answer is correct using the definition of the limit of a function. (Do not rely on

any other results about limits from class.)

6. (12 points) Let f : A → B be any function and let D1, D2 be any non-empty subsets of B.

Proof or counterexample:

f−1(D1 ∪D2) = f−1(D1) ∪ f−1(D2)

7. (14 points) (a) State the definition of compactness (using the notion of open covers)

(b) State the Heine-Borel theorem

(c) Complete the following proof of the Bolzano-Weierstrass theorem (which states that every

bounded, infinite subset of R has an accumulation point):

Page 2

Proof by contradiction: Let S be a bounded set with infinitely many points, and such that S

has no accumulation points. Thus, S is closed, since it trivially contains all of its accumulation

points. So by the Heine-Borel theorem, S is compact.

Since S has no accumulation points, every point x ∈ S has a neighborhood Nx with Nx ∩ S =

{x}. Thus the set {Nx, x ∈ S} are an open cover. But...

8. (12 points) Find the limit of the sequence {sn} defined according to the relations s1 = 1 and

sn+1 =

1

5(sn + 7) for all n ≥ 1. Prove your answer is correct using the Monotone Convergence

Theorem.

Page 3