Global mark: / = %

MTH2140/MTH3140 Real Analysis – Project 2 (2015)

Due date: 12noon on Friday 29 May

Submit this assignment before the due date in your demonstrator’s assignment box, located on the ground floor of Building 28. Please refer to the unit guide for general assessment criteria (in particular the writing requirement, e.g. “explain what you are doing, use complete English sentences...”) and for submitting late assignments. Write your answers in the provided spaces on this sheet.

Name: Student ID:

Tutorial’s time: Level: MTH2140 2

MTH3140

5 marks will be allocated to the display of communication skills. This includes the use of proper English 2 sentences and the overall presentation of the assignment (no rough working or striked out parts).

Exercise 1. (3+5+7=15 marks) For x ? [0,p/2] we define

Marks: /5

.

1. Find the pointwise limit of (fn), i.e. fund the function f : [0,p/2] ? R such that fn(x) ? f(x) for all x ? [0,p/2].

Solution:

Solution:

Marks: /5

3. Prove that for all a ? (0,p/2], fn ? f uniformly on [a,p/2] (Hint: sin is non-decreasing on [a,p/2], so sin(x) = sin(a) if x ? [a,p/2]). Solution:

Exercise 2. (3+3+6+8=20 marks)

1. Let f : [0,8) ? R be continuous such that limx?+8 f(x) = 0. Prove that f is bounded on [0,8).

.

Solution:

Marks: /3

3. Deduce that Pne-na converges for any a 0.

Solution:

Marks: /6

4. Prove that the series Pe-nx+cos(nx) is defined, continuous and differentiable (with a continuous derivative) on (a,8) for any a 0.

Exercise 3. (MTH3140 only) (4+2+4=10 marks) We want to prove the following proposition, mentioned in the lecture notes.

Proposition 1 Let (fn) be defined and continuous on an interval [a,b], and differentiable on (a,b). Let c ? [a,b]. Assume that (fn(c)) converges and that converges uniformly on (a,b). Then (fn) converges uniformly on [a,b].

1. Let (fn) be a sequence of functions that are continuous on [a,b] and differentiable on (a,b). Use the Lipschitz estimate to prove that |fn(x)-fp(x)-(fn(c)-fp(c))| = |b-a|supy?(a,b) |fn0 (y)-fp0(y)| for all x ? [a,b] and all n,p ? N (make explicit the function on which you use the Lipschitz estimate).

Solution:

Marks: /4

2. Deduce that

|fn(x) - fp(x)| = |fn(c) - fp(c)| + |b - a| sup |fn0 (y) - fp0(y)|. (1)

y?(a,b)

Solution:

Solution:

MTH2140/MTH3140 Real Analysis – Project 2 (2015)

Due date: 12noon on Friday 29 May

Submit this assignment before the due date in your demonstrator’s assignment box, located on the ground floor of Building 28. Please refer to the unit guide for general assessment criteria (in particular the writing requirement, e.g. “explain what you are doing, use complete English sentences...”) and for submitting late assignments. Write your answers in the provided spaces on this sheet.

Name: Student ID:

Tutorial’s time: Level: MTH2140 2

MTH3140

5 marks will be allocated to the display of communication skills. This includes the use of proper English 2 sentences and the overall presentation of the assignment (no rough working or striked out parts).

Exercise 1. (3+5+7=15 marks) For x ? [0,p/2] we define

Marks: /5

.

1. Find the pointwise limit of (fn), i.e. fund the function f : [0,p/2] ? R such that fn(x) ? f(x) for all x ? [0,p/2].

Solution:

Solution:

Marks: /5

3. Prove that for all a ? (0,p/2], fn ? f uniformly on [a,p/2] (Hint: sin is non-decreasing on [a,p/2], so sin(x) = sin(a) if x ? [a,p/2]). Solution:

Exercise 2. (3+3+6+8=20 marks)

1. Let f : [0,8) ? R be continuous such that limx?+8 f(x) = 0. Prove that f is bounded on [0,8).

.

Solution:

Marks: /3

3. Deduce that Pne-na converges for any a 0.

Solution:

Marks: /6

4. Prove that the series Pe-nx+cos(nx) is defined, continuous and differentiable (with a continuous derivative) on (a,8) for any a 0.

Exercise 3. (MTH3140 only) (4+2+4=10 marks) We want to prove the following proposition, mentioned in the lecture notes.

Proposition 1 Let (fn) be defined and continuous on an interval [a,b], and differentiable on (a,b). Let c ? [a,b]. Assume that (fn(c)) converges and that converges uniformly on (a,b). Then (fn) converges uniformly on [a,b].

1. Let (fn) be a sequence of functions that are continuous on [a,b] and differentiable on (a,b). Use the Lipschitz estimate to prove that |fn(x)-fp(x)-(fn(c)-fp(c))| = |b-a|supy?(a,b) |fn0 (y)-fp0(y)| for all x ? [a,b] and all n,p ? N (make explicit the function on which you use the Lipschitz estimate).

Solution:

Marks: /4

2. Deduce that

|fn(x) - fp(x)| = |fn(c) - fp(c)| + |b - a| sup |fn0 (y) - fp0(y)|. (1)

y?(a,b)

Solution:

Solution:

Assessment: 02- Report / Project / Role PlayAssessment Description:This is a Holistic assessment, developed to satisfy the Performance Elements along with the Performance Evidences of the unit. The assessment...Mr Simon has spent 12 years continuous service as a line manager in a retail organisation. In the past three years he has twice been considered for promotion via a formal interview process and was unsuccessful....MSc Project Proposal Do not add any front sheets or title pages. Read the notes in italics and remove before submitting. You should also delete this paragraph of text. Minimum page limit of 3 and maximum...Adept owl scenario procedure instructor 2Please use examples from literature to support your ideas in your discussion.LOI, 2, 3 & 4 (1500 words)Consideryou might want to reflect on your experiences as a child and how these influence your...CUC107 Assessment 1: Cultural Mind MapValue: 15%Time: This task should take about 4-5 hours. Word count (250-300) images will varyDue: Monday, 11:59pm (CST), Week 5Submission:• in PDF format at the 'Assessment...Hi This is 800 words assignment based on case study and its the part B i.e. scope and time management plans Part B – Scope and Time Management Plans In this subject, you will be creating key components...**Show All Questions**