## Description

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?