MATH5301 Elementary Analysis. Homework 7.
7.1
Provide an examples of the sets A, B ⊂ R
2
such that
(a) A and B are connected, but A ∪ B is not.
(b) A and B are connected, but A ∩ B is not.
(c) A and B are not connected, but A ∪ B is connected.
(d) A and B are not connected, but A ∩ B is connected.
(e) A and B are not connected, but A \ B is connected.
7.2
(a) Prove that every monotone bounded sequence in R converge.
(b) Provide an example of the set A ∈ R having exactly four limit points.
(c) Provide an example of a sequence {an}, such that every point of the interval [2019, 2021] is a limit point
of it.
7.3
(a) Provide an example of a sequence {an} such that an diverges, but limn→∞
(an − a2n) = 0
(b) Provide an example of two sequences {an} and {bn} such that
(lim inf
n→∞
an+lim inf
n→∞
bn) < lim inf
n→∞
(an+bn) < (lim inf
n→∞
an+lim sup
n→∞
bn) < lim sup
n→∞
(an+bn) < (lim sup
n→∞
an+lim sup
n→∞
bn)
7.4
Show the equivalence of the norms k · k1, k · k2, k · kp, p > 1 and k · k∞ on R
n
7.5
Are there any open sets A and B4 in R
2
such that d(A, B) = 0 but A ∩ B = ∅?
7.6
Let B([0, 1]) denote the set of all bounded functions from [0, 1] to R. Define the metric on B[0, 1] as d(f, g) =
sup
x∈[0,1]
|f(x) − g(x)|.
(a) Show that this is indeed a metric.
(b) Prove that the space (B([0, 1]), d) is complete metric space.
(c) Is the unit ball B1(0) = {f(x) | d(f, 0) 6 1} compact?
