Overview
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 de nes 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 de ned 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 di erent
elements of this sequence?)
Remark on notation. If you haven't seen the symbol \:=" before, it just means we are de ning a notation
for something. For instance in this last problem, we de ne 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