EXAM I: PRACTICE SHEET
(1) Let S and T be two bounded subsets of R. Define
S + T = {s + t : s ∈ S and t ∈ T }.
Prove that
(a) S + T is a bounded subset of R.
(b) sup(S + T ) = sup(S) + sup(T ).
(c) inf(S + T ) = inf(S) + inf(T ).
(2) (a) Find with justification
o
n1
: n = 1, 2, . . . .
inf
n
(b) Find with justification
n 4n + 1
o
sup
: n = 1, 2, . . . .
5n + 3
(c) Let a > 0, find with justification sup{x : x ∈ R, x > 0, x2 < a}.
(3) Let b > 1 be a real number. Define the sequence {an } as follows: a1 = 1
and
an + b
for all integers n.
an+1 =
2
(a) Show that 0 < an < b for all n.
(b) Find sup{an : n = 1, 2, . . .}. Justify your answer.
(c) Find inf{an : n = 1, 2, . . .}. Justify your answer.
(4) Prove that an open interval in R is an open set and that a closed interval
is a closed set.
(5) Determine all the limit points of the following sets in R and determine
whether the sets are open or closed (or neither).
(a) All integers.
(b) The interval (a, b].
(c) All numbers of the form n1 , where n = 1, 2, 3, . . .
(d) All rational numbers.
(e) All numbers of the form 2−n + 5−m , where m, n = 1, 2, . . .
1
(f) All numbers of the form (−1)n + m
, where m, n = 1, 2, . . .
(g) All numbers of the form
(h) All numbers of the form
1
1
n + m where m, n = 1, 2, . . .
(−1)n
1+(1/n) , where n = 1, 2, . . .
(6) Let S be a subset of Rn . Show that
(a) S 0 is a closed set.
(b) If S ⊂ T, then S 0 ⊂ T 0 .
(c) (S ∪ T )0 = S 0 ∪ T 0 .
1
2
EXAM I: PRACTICE SHEET
(d) (S)0 = S 0 .
(e) S is a closed set.
(f) S is a closed set if and only if S = S.
(7) Show that every closed subset of R is an intersection of countably many
open sets. (Hint: Say A ⊂ R is closed subset. For each n consider On =
∪a∈A (a − n1 , a + n1 .) Show that A = ∩∞
n=1 On and conclude.)
(8) In this problem you may use without a proof the fact that Q is dense in R
with the usual metric.
(a) Show that A = {a + bi : a, b ∈ Q} is dense in C with the usual metric
on C.
(b) Show that Qn is dense in Rn with the usual metric on Rn .
(c) Let A be as in part (a). Show that An is dense in Cn with the usual
metric on Cn .
(9) Let
n
o
`∞ = (a1 , a2 , . . .) : an ∈ R & sup{|an | : n ∈ N} < ∞ .
We use componentwise addition and scalar multiplication on `∞ . For any
(an ) ∈ `∞ , define
(an ) = sup{|an | : n ∈ N}.
∞
Define
d (an ), (bn ) = (an − bn )∞
for any two sequences (an ) and
(bn ).
(a) Show that d (an ), (bn ) is a finite number for any two (an ) and (bn )
in `∞ .
(b) Show that `∞ , d is a metric
space.
(c) (Bonus Problem) Is `∞ , d a separable space? (A metric space X is
called separable if there is a countable and dense subset A ⊂ X, e.g. R
with the usual metric is separable. )
(10) For any m ∈ N ∪ {0} let
Xm = {p(z) =
m
X
an z n : z ∈ C, |z| = 1, an ∈ C},
n=0
S∞
and let X = m=0 Xm .
(a) For any p(z) ∈ X show that the set {|p(z)| : |z| = 1} is a bounded set
and conclude that
sup{|p(z)| : |z| = 1}
exists.
(b) For any p(z), q(z) ∈ X define d p(z), q(z) = kp(z) − q(z)k∞ where
kr(z)k∞ = sup{|r(z)| : |z| = 1} for any r(z) ∈ X.
Show that (X, d) is a metric space.
(c) Is (Xm , d) a separable metric space? How about (X, d)? (Hint: Problem 8(c) is useful.)
EXAM I: PRACTICE SHEET
3
(11) For any m ∈ N, let
Xm = {(zn ) : zn ∈ C and zn = 0 ∀ n > m},
S∞
P
and let X = m=1 Xm . For any (zn ) ∈ X define k(zn )k1 = n |zn |, note
that this is a finite sum.
(a) For any (zn ), (wn ) ∈ X define d (zn ), (wn ) = (zn ) − (wn )1 . Show
that (X, d) is a metric space.
(b) Is (X, d) separable.
(iii) For each m ∈ N define z(m) ∈ X as follows
(
1
if 1 ≤ n ≤ m
n
z(m) n = 2
.
0
otherwise
Show that kz(m)k1 ≤ 1 for all m ∈ N.
(c) Let z(m) be defined as above, show that d z(m), z(m0 ) ≤
(d) Does {z(m) : m ∈ N} have a limit point in X?
1
.
2min{m,m0 }
1. Bonus problems
(12) Let p be a prime number. For any integer m ∈ Z define
(
power of p in the prime factorization of m if m 6= 0
νp (m) =
.
∞
if m = 0
Define the following norm on Q.
(
νp (n)−νp (m)
m
= p
n p
0
if
if
m
n
m
n
6= 0
.
=0
0
m
m
(a) Show that this definition is well defined, i.e. if m
n = n0 , then | n |p =
m0
| n0 | p .
(b) For any two r, s ∈ Q define dp (r, s) = |r − s|p . Show that (Q, dp ) is a
metric space.
(c) Show that Z is a bounded subset of Q with respect to this metric.
(d) Does {pn : n ∈ N} have a limit point in Q?
(13) Let A and B be two sets. Assume that there are 1-1 maps f : A → B and
g : B → A. Prove that there exists a one to one correspondence between A
and B.
S∞
Hint: Let D = n=0 (g ◦ f )n A − g(B) .
(
f (a)
a∈D
h(a) =
.
−1
g (a) a ∈ A \ D
Show that h is well-defined, 1-1, and onto.