Real Analysis - Project
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].
2. 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).
2. Deduce that Pne-na converges for any a 0.
3. 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. (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)
y?(a,b).