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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
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
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].
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).
3. Deduce that Pne-na converges for any a 0.
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).
2. Deduce that
|fn(x) - fp(x)| = |fn(c) - fp(c)| + |b - a| sup |fn0 (y) - fp0(y)|. (1)