MATH 118-A FALL 2021
PRACTICE PROBLEMS FINAL
1) Let (X, d) be a metric space. Let A = {an : n ∈ Z+} be a Cauchy sequence
in X. Using only the definition of Cauchy sequence prove that A is bounded.
2) (a) Let (X, d) be a metric space. Using only the notion of open set recall the
definition of K X being compact.
(b) In R with the usual distance, i.e. x, y ∈ R d(x, y) = |x y|. Prove using
just the definition of compactness that A = (0, 1] is NOT compact.
3) Let K be a compact sub-set on a metric space (X, d). Prove that if A ? K
is closed, then A is compact.
4) In each case give an example of A R and f : R→ R continuous such that
(i) A compact with f?1(A) no compact.
(ii) A connected with f?1(A) no connected.
(iii) A open with f(A) not open.
(iv) A closed with f(A) not closed.
5) Prove or give a counter-example of the following statements
(i) (interiorA) ? interior(Aˉ).
(ii) interior(Aˉ) ? (interiorA).
(iii) interior(A ∪B) = interiorA ∪ interiorB.
(iv) A ∪B = Aˉ ∪ Bˉ.
6) Consider Rn with the usual distance. Using only the definition of compact
set prove that if K ? Rn is compact, then K is closed and bounded.
7) Let A = {xk : k ∈ Z+} ? R be an non-increasing sequence
i.e. ? k ∈ Z+, xk ≥ xk+1. Prove that if A is bounded below, then the sequence
converges.
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2 MATH 118-A FALL 2021 PRACTICE PROBLEMS FINAL
8) Prove or give a counter-example of the following statements :
(i) Let (X, d) be a metric space and A, B ? X , then A ∩B = A ∩B.
(ii) Let (X, d) be a metric space and An, n = 1, 2, 3... a sequence of open set,
then
∩∞n=1An is open.
(iii) Let (X, d) be a metric space and A, B ? X , then
interior(A ∩B) = interior(A) ∩ interior(B).
(iv) If f : Rn → Rn continuous and C closed and bounded, then f(C) is closed.
9) Let {ak : k ∈ Z+} ? R be a non-increasing sequence
i.e. ? k ∈ Z+, ak ≥ ak+1 such that
lim
k→∞
ak = 0.
Prove that the series ∞∑
k=1
(?1)k+1ak converges.
10) Let (X, d) be a metric space. Let {Aα}α∈I be a family of connected sub-sets
of X. Prove :
if ∩α∈I Aα 6= ?, then ∪α∈I Aα is connected.
12) Prove that
(i) f : R→ R with f(x) = sin(x) is uniformly continuous.
(ii) h : (1/2,∞)→ R with h(x) = 1/x is uniformly continuous.
(iii) g : (0, 1]→ R with g(x) = 1/x is not uniformly continuous.
(iv) z : R→ R with z(x) = 1/(1 + x2) is uniformly continuous.
13) Let Θ ? Rn be an open set and K be a compact set such that K ? Θ. Prove
that there exists a set D such that :
(i) D is compact,
(ii) K ? (D)o = interior(D)
(iii) D ? Θ.
14) Let A ? R be a closed set such that Q ∩ [0, 1] ? A. Prove that [0, 1] ? A.
15) Let fj : Rn → R, j = 1, .., n be defined as fj(x1, .., xn) = xj . Prove that fj
is uniformly continuous in Rn.
MATH 118-A FALL 2021 PRACTICE PROBLEMS FINAL 3
16) Let F : R4 → R be defined as F (x1, x2, x3, x4) = x1x4 ? x2x3.
(a) Prove that F is continuous.
(b) Prove that F?1({0}) is a closed set of R4.
(c) Prove that F?1(R? {0}) is a open set of R4.
(d) By identifying R4 with
M2×2(R) = {
(
x1 x2
x3 x4
)
: x1, x2, x3, x4 ∈ R}.
what can you conclude from part (b) and part (c) ?
*17) Let (X, d) be a metric space.
(a) Assuming that X is compact, prove that it does not exist f : X → X
continuous such that
(Z) ?x, y ∈ X d(f(x), f(y)) d(x, y) if x 6= y.
(b) Give an example of f : R→ R continuous satisfying (Z).
*18) Let A = {(x, sin(1/x)) ∈ R2 : x ∈ (0, 1]}. Find Aˉ.
19) In a metric space (X, d) the boundary of a set A ? X is defined as
?(A) = bddry(A) = A ∩Ac.
Prove : (i) ?(A) = ?(Ac),
(ii) ?(A) = A? interior(A),
(iii) ?(A) ∩ interior(A) = ?,
(iv) A = ?(A) ∪ interior(A),
(v) X = interior(A) ∪ ?(A) ∪ interior(Ac).
20) Let (X, dX), (Y, dY ) be two metric spaces. Let f : X → Y be a continuous
functions. Prove that for any A ? X
(2) f(Aˉ) ? (f(A))
Give examples where the strictly inequality (2) holds.
21) In R2 let
A1 = Q×Q and A2 = Qc ×Qc,
Prove that R2 ?A1 is connected and A1, A2 are not connected.
4 MATH 118-A FALL 2021 PRACTICE PROBLEMS FINAL
22) Let f : (0, 1)→ R be defined as
f(x) =
{
1/q, x = p/q p, q relative prime
0, x irrational.
Where is f continuous ?
23) Let f : R→ R be a monotonic non-decreasing function, i.e.
?x, y ∈ R x < y implies f(x) ≤ f(y).
Prove that set
{ z ∈ R : f discontinuous at z}
is at most countable.
24) Consider Rn with the usual distance, Let f : Rn → R be a continuous
function and K ? Rn be a compact and connected set. Describe f(K).
25) Let (X, d) be a COMPLETE metric space. Let φ : X → X be a contraction,
i.e.
? θ ∈ (0, 1) ?x, y ∈ X d(φ(x), φ(y)) ≤ θ d(x, y).
Prove that there exists a unique x? ∈ X such that φ(x?) = x? (fixed point).
HINT. Let x0 ∈ X any point and x1 = φ(x0) and xn+1 = φ(xn), n ∈ N. Prove :
d(xn, xn+k) ≤ θk d(x0, x1).
Prove that (xn)
∞
n=1 is a Cauchy sequence.
26) Let (X, d) be a metric space. If K ? X is compact, then every infinite subset
A of K has a limit point in K.
27) Let (X, d) be a metric space and (an)
∞
n=1 be a sequence of elements of X.
MAKE precise the following statement : an converges to L ∈ X if nd only if every
subsequence of (an)
∞
n=1 has a sub-sub-sequence which converges to L.
28) Prove that given any λ > 0 there exists an increasing sequence of integers
n1 < n2 < ..... < nk < nk+1 < ..... nj ∈ N
such that ∞∑
k=1
1
nk
= λ.
29) Let (X, dX) and (Y, dY ) be two metric spaces. Prove that if f : X → Y is
uniformly continuous, the if sends Cauchy sequence in X to Cauchy sequence in Y .
Prove that if f is just continuous the same results fails.
MATH 118-A FALL 2021 PRACTICE PROBLEMS FINAL 5
30) Prove that for any x ∈ R the series
∞∑
n=1
sin(nx)
n
converges.
HINT : uses problems 5-6 of HW#6.
31) Give an example of a function f : (0, 1)→ R bounded and continuous which
is not uniformly continuous.
32) Let f : [0, 1]→ R be a continuous function such that f(0) < 0 and f(1) > 1.
Show that there exists c ∈ (0, 1) such that f(c) = c3.
33) Give an example of a function f : (0, 1)→ R continuous which does not send
Cauchy sequences into Cauchy sequences.
34) Let (X, dX) and (Y, dY ) be two metric spaces. Prove that if f : X → Y is
uniformly continuous and (an)
∞
n=1 is a Cauchy sequence in X , Then (f(an))
∞
n=1 is
a Cauchy sequence in Y .
34) Let (X, dX) and (Y, dY ) be two metric spaces. Prove that f : X → Y is
continuous at x0 ∈ X if and only if for any sequence (xn)∞n=0 in X
if lim
n→∞xn = x0, then limn→∞ f(xn) = f(x0).
35) Prove that f : R→ R with f(x) = x2 is not uniformly continuos.
36) Prove that f : R→ R with f(x) = sin(x) is uniformly continuos.
37) Let (X, dX) and (Y, dY ) be two metric spaces. Prove that if f : X → Y is
continuous. Let A ? Y
(i) What is the relation between the sets f?1(A) and f?1(A) ?
(ii) Give an example where f?1(A) 6= f?1(A).
(iii) What is the relation between the sets int{f?1(A)} and f?1(int{A}) ?
(ii) Give an example where int{f?1(A)} 6= f?1(int{A}).