Real Analysis I




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

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