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UW-Madison MATH 521 - TermTestII-sample

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Math 521, Spring 2014, Term Test II (Sample)Analysis IDate: Friday, April 4Name (printed):UW Student ID Number:Instructions1. Fill out this cover page.2. Answer questions in the space pro-vided, using backs of pages for over-flow and rough work.3. Show all work required to obtainyour answers.4. Unless otherwise stated, you mayuse any theorem or result derived inclass.FOR EXAMINERS’ USE ONLYPage Mark2 /∞3 /∞4 /∞5 /∞Total /∞Math 521, Term Test II, Spring 2014 Page 2 of 5 Name:1. Definitions and Theorems:(a) State the Cauchy-Schwarz Inequality.(b) Suppose x, y ∈ Rn. State the definitions of the following:i. The norm kxk.ii. The dot product x ·y.iii. The Euclidean metric d2(x, y) = kx − yk.(c) Consider a pairing (X, d) where X is some set and d : X × X 7→ R. State therequirements on d for (X, d) to be a metric space.(d) Suppose (X, d) is a metric space and S ⊆ X. State the definitions of the following:i. The open ball Br(x) of radius r > 0 around x ∈ X.ii. A neighborhood N(x) of x ∈ X.iii. A family {Sα}, α ∈ A, is a open cover of S.iv. x ∈ S is an interior point of S.v. x ∈ X is a limit point of S.vi. x ∈ X is a boundary point of S.vii. x ∈ S is an isolated point of S.viii. So⊆ X is the interior of S.ix. S ⊆ X is the closure of S.x. S is an open set.xi. S is a closed set.xii. S is a compact set.(e) State the Heine-Borel Theorem.(f) State the Bolzano-Weierstrass Theorem.2. Metric Spaces(a) Consider the sets X = {A, B, C, D} where the distances between elements aredefined as follows:ABCD321365(i)(ii)ABCD221221For example, in (i) we define d(A, B) = d(B, A) = 2. (We also allow d(A, A) = 0,etc.) Do the pairings (X, d) represent metric spaces for (i) and (ii)? Why or whynot?2Math 521, Term Test II, Spring 2014 Page 3 of 5 Name:(b) Show that (R, d) with the following functions d : R ×R 7→ R are not metric spaces:i. d(x, y) = x2− y2ii. d(x, y) =|x| − |y|iii. d(x, y) = |x − y|1pfor any p > 0, p 6= 1 (Hint: Consider p ∈ (0, 1) and p > 1separately)3. Topology(a) Let (X, d) be a metric space and S ⊆ X. Prove the following:i. (So)o= Soii. S = Siii. Sc= (So)civ. ∂S ⊆ S0∪ (Sc)0v. (∂S)c= So∪ (Sc)ovi. If S is open, then Scis closed.vii. If S is closed, then Scis open.(b) Prove that Br(x) is an open set (for all r > 0 and all metric spaces (X, d)).(c) Prove that if A, B ⊆ X are open sets, then A ∩ B is an open set.(d) Prove that if A, B ⊆ X are closed sets, then A ∪ B is a closed set.(e) Give an example of an infinite family {Sn}, n ∈ N, such thati. All Snare open but∞\n=1Snis non-empty and closed.ii. All Snare closed but∞[n=1Snis non-empty and open.4. Connected Sets(a) Determine whether the following sets S ⊆ X are connected. If they are, explainwhy. If they are not, find nonempty separated sets A, B ⊂ X so that A ∪B = S.i. S = {x ∈ R | x > 0}, in (R, d), d(x, y) = |x −y|ii. S = {x ∈ R | 2 < x2< 3}, in (R, d) , d(x, y) = |x −y|iii. S =x ∈ R2| 0 < kxk ≤ 1, in (R2, d2)iv. S = Q in (Q, d), d(x, y) = |x − y|(b) Consider the metric space (X, d) with d(x, y) = 0 if x = y and d(x, y) = 1 forx 6= y. Prove that every subset S ⊆ X with at least two elements is disconnected.3Math 521, Term Test II, Spring 2014 Page 4 of 5 Name:5. Compact Sets(a) Consider the metric space (R, d) with d(x, y) = |x − y|. For the following sets,construct an open cover which does not have a finite subcover.i. A = {x ∈ R | x ≥ 0}ii. B =(−1)n1n∈ R | n ∈ Niii. C = [−1, 0) ∪ (0, 1](b) Consider (R, d) as before and show directly that every open cover of the followingset has a finite subcover (i.e. you may not use the Heine-Borel Theorem!):i. A =nn2+ 1| n ∈ N∪ {0}ii. B =(−1)n1 −1n∈ R | n ∈ N∪ {−1} ∪ {1}(c) Let (X, d) be a metric space. Suppose that A ⊆ X is compact and B ⊆ X is closed.Prove that A ∩ B is compact. (Note: Since (X, d) is a general metric space, not(Rn, d), we may not use the Heine-Borel Theorem!)6. Sequences(a) For the following sequences {an}, determine the limit as n → ∞ and then provethe convergence using the definition:i. an=√n2− 1 − nii. an=2n2n2+ 4iii. an=n√n2+ 1(b) For the following recursively defined sequences {an}, determine the limit as n → ∞and then prove the convergence using some result from class:i. an+1= −12an− 2ii. an+1=√2 + an, a1= 1(c) Prove that every sequence {an} ⊂ R with an+1= can+d (where c, d ∈ R) convergesin R if 0 ≤ c < 1. (Hint: Find the limit value and then relate |an+1−L| to |an−L|.)(d) Use the definition to prove that, for a sequence {an} with an≥ 0 for n ∈ N, if {an}converges to a then {√an} converges to√a.4Math 521, Term Test II, Spring 2014 Page 5 of 5 Name:THIS PAGE IS FOR ROUGH


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