Find sets A ( B in (R; j:j) such that A is open in R and relatively closed in B.

Problem 1. Find sets A ( B in (R; j:j) such that A is open in R and relatively closed in B.

Problem 2. Let (xn)1 n=1 be a sequence in a metric space (X; d).

(a) Suppose (xn)1 n=1 is a Cauchy sequence. Show that fxn j 1 n < 1g is a bounded set.

(b) Suppose (xn)1 n=1 is a Cauchy sequence, and that there is a subsequence (xnj )1j

=1 which converges to

some x 2 X. Show that also limn!1 xn = x.

(c) We say that x 2 X is a limit point of the sequence (xn)1 n=1 if

8″ > 0; 8N 2 N; 9n N : d(xn; x) < “:

Show that x is a limit point of (xn)1 n=1 if and only if there exists a subsequence (xnj )1j

=1 which

converges to x.

Problem 3. Exercises 1.5.12, 1.5.13, 1.5.14 from the book.

Problem 4. Let (X; d) be a totally bounded metric space, and E X a subset. Show that also (E; dE)

is totally bounded.

Problem 5 (Fancy example of a closed and bounded set which is not compact). Let `1 P (R) := f(an)n2N j

n2N janj < 1g.

(a) Show that d((an)n; (bn)n) :=

P

n2N jan ? bnj denes a metric on `1(R).

(b) Consider the so-called unit ball B := f(an)n2N 2 `1(R) j

P

n2N janj 1g. Show that this unit ball is

closed and bounded, but not compact.

Hint for non-compactness: Consider the sequence (a(m))1m

=1 in B dened by

(

a(m)

m = 1;

a(m)

n = 0; if n 6= m;

and show that it does not have a convergent subsequence. (What is the distance between two dierent

elements of this sequence?)

Remark on notation. If you haven’t seen the symbol :=” before, it just means we are dening a notation

for something. For instance in this last problem, we dene the notation `1(R)” to stand for the set of all

sequences (an)n2N satisfying

P

n2N janj < 1. Some people also write def =”.

1

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