MATH 118代写、R程序设计代写

日期：2022-12-08 09:48

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}).