## Transcribed Text

Real Analysis { Project
Exercise 1. (3+5+7=15 marks) For x 2 [0; =2] we dene
fn(x) =
nx
1 + n sin(x)
:
1. Find the pointwise limit of (fn), i.e. fund the function f : [0; =2] ! R such that fn(x) ! f(x) for all x 2 [0; =2].
Solution:
2. Prove that fn does not converge uniformly to f on [0; =2].
3. Prove that for all a 2 (0; =2], fn ! f uniformly on [a; =2] (Hint: sin is non-decreasing on [a; =2], so sin(x) sin(a)
if x 2 [a; =2]).
Solution:
Exercise 2. (3+3+6+8=20 marks)
1. Let f : [0;1) ! R be continuous such that limx!+1 f(x) = 0. Prove that f is bounded on [0;1).
Solution:
2. Deduce that for any a > 0 there exists C 2 R such that for all n 1
nena Cena
2 :
P
3. Deduce that nena converges for any a > 0.
Solution:
4. Prove that the series
P
enx+cos(nx) is, continuous and differentiable (with a continuous derivative) on (a;1)
for any a > 0.
Solution:
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 jfn(x)fp(x)(fn(c)fp(c))j jbaj supy2(a;b) jf0n
(y)fp 0 (y)j for all x 2 [a; b] and all n; p 2 N (make
explicit the function on which you use the Lipschitz estimate).
Solution:
2. Deduce that
jfn(x) fp(x) jfn(c)fp(c)j + jbaj sup
y2(a;b)
n p jf0 (y f0 (y)j:
Solution:
3. Prove Proposition 1 (hint: uniform Cauchy criterion for sequences of functions).
Solution:
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 and continuous on an interval [a; b], and differentiable on (a; b). Let
c 2 [a; b]. Assume that (fn(c)) converges and that (f0n
) converges uniformly on (a; b). Then (fn) converges
uniformly on [a; b].

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.