B.Ed(Arts)

SMA 301/TMA 301: REAL ANALYSIS IC.A.T]1

INSTRUCTIONS:

Answer anyTWOQuestions.

Question 1

a).[6 marks] Let (X,d) be a metric space. Show that for allx,y,z,p∈X, we have|d(x,y)−d(z,p)|≤d(x,z) +d(y,p).

(b).[5 marks] LetXbe a non-empty set andf:X−→Rbe a one-to-one function. Defined:X×X−→R+byd(x,y) =|f(x)−f(y)|, for allx,y∈X. Show thatdis a metric onX.(c).[4 marks] Findint(A) andAof the following subsets of (R,d), wheredis the usual metric onR.(i).A=Q∩[0,4].(ii).A={1n:n∈N}.

Question 2

(a).(i).[6 marks] Let (X,d) be a metric space and suppose thatAis an open subset ofX. Prove thatA∩B⊆A∩B.Show by a counterexample that this result need not hold ifAis not open.(ii).[6 marks] Letdbe the metric onX= (0,∞) defined byd(x,y) =|1x−1y|, for allx,y∈X. Show thatdisequivalent to the usual metricρ(x,y) =|x−y|onX.

(b).[3 marks] Give an example of a metric space (X,d) and a non-empty subsetAsuch that every point inAis alimit pointAbutint(A) =∅.

Question 3

(a). Suppose (Xi,di) are metric spaces (i= 1,2,…,n) and thatx= (x1,x2,…,xn) andy= (y1,y2,…,yn) arepoints inX=X1×X2×…×Xn.(i).[9 marks] Prove that each of the following functions is a metric onX:ρ1(x,y) =√√√√n∑i=1di(xi,yi);ρ2(x,y) =n∑i=1di(xi,yi);ρ3(x,y) = max{di(xi,yi) :i= 1,2,…,n}.(ii).[3 marks] Prove that if eachXi=Rand eachdiis the usual metric onR, thenρ1,ρ2andρ3are just theusual metric, the taxicab metric and the max metric onRn, respectively.

(b).[3 marks] LetdEanddDdenote the Euclidean and discrete metrics, respectively. Draw the following openballsB(x,r)(i).B(−1,4)⊂(R,dE).(ii).B((1,1),1)⊂(R2,dD).(iii).B((1,1),2)⊂{(x,y)∈R2: 0≤x≤1,0≤y≤1}⊂(R2,dD).1

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more