Analysis I - Faculty.jacobs-university.de - Jacobs University

Jacobs University
School of Engineering and Science
Peter Oswald, Alex Sava, Eugenia Rosu
Homework Problems
Analysis I
Fall Term 2009
Problem Set 6
6.1. More on Cantor-type sets
The construction of the Cantor set is modified as follows. Fix a number α ∈ (0, 1). Start with
the interval [0, 1], and call it C1. To get C2, remove the central portion ( 1−α 1−α , ) (an open
2 2
interval of length α) from C1. I.e., C2 is the union of 2 closed intervals (of equal length). Now
do the following recursive procedure: After you have constructed Cn as the union of 2n−1 closed, and pairwise disjoint intervals of equal length, remove from each of these intervals
an open interval of length αn symmetrically located about its midpoint. Show the following
facts:
a) (2 pts) For α > 1/3, the procedure breaks down after finitely many steps.
b) (3 pts) For α ∈ (0, 1/3], the resulting Cantor-type set C = ∩Cn is closed, and non-empty.
c) (5 pts) For α ∈ (0, 1/3], C is perfect but does not contain any closed interval of positive
length (such C ⊂ IR are called totally disconnected).
6.2. Useful tool for computing limits: The Stolz-Cesaro Lemma
a) (7 pts) Suppose an and bn are two sequences of real numbers such that bn is positive,
strictly increasing, and unbounded (i.e., bn → +∞). Then :
an
lim
n→∞ bn
an+1 − an
= lim
n→∞ bn+1 − bn
if the limit on the right-hand side exists.
Hint: Just use the ɛ-n0 definition of convergence.
Comment: This is sometimes called the discrete L’Hospital rule.
b) (3 pts) Compute the following limit:
6.3. Weierstrass Theorem - Sequences in IR
1
lim
n→∞
k + 2k + . . . + nk nk+1 a) (4 pts) Prove that from any sequence of real numbers we can extract a monotone
subsequence (increasing or decreasing, not necessarily both!).
b) (3 pts) Prove that if a sequence is bounded above, and monotonically increasing then
it is convergent.
c) Show that an → 0 and (bn)n∈IN bounded implies anbn → 0.

6.4. About the set of limit poits (= subsequential limits)
a) (7 pts) Let (pn)n∈IN be a sequence in a metric space (X, d). Then the set of all limit points
of (pn)n∈IN is closed.
b) (3 pts) What do you think: Can any closed set E ⊂ IR be realized as the set of limit
points of a certain sequence in IR? Could such a statement be true for any metric space?
6.5. About upper and lower limits
(10 pts) Prove that: lim sup(an+bn) ≤ lim sup an+lim sup bn. Try to show a similar property
for lim inf.
6.6. Bonus problem
(10 pts) Consider the two recursively defined sequences
an+1 = an
2
5
+ , bn+1 = bn + 2 −
an
b2n , n ≥ 1,
5
with starting values a1 = b1 = 2 in IR. Show that both sequences converge to the same limit
L, and try to answer the question which one converges faster.
Hint: First assume convergence, and find the limit value. After this, investigate the behavior
of the respective error sequences ɛn = an − L and δn = bn − L.
Due Date: Thursday Oct. 29, in class.