5

# 1- Complete the following proof. Theorem Let S ER: Let p â‚¬ R The following two statements are equitlent: (i) Ve > Ve(p)o (S| {p}) "onemptu: (ii) P= l...

## Question

###### 1- Complete the following proof. Theorem Let S ER: Let p â‚¬ R The following two statements are equitlent: (i) Ve > Ve(p)o (S| {p}) "onemptu: (ii) P= lim dn for some sequence (an) contained in $| {p} (Recall that if any of the above equivalent conditions holds, then we say that is a limit poinf of$.) Proof: (ii) Let â‚¬ be given_ Qur goal , is to show that Ve(p)n(S| {p}) is nonempty: We have Uj p JN â‚¬N Vn >N dn â‚¬ Velp) In particular; aN+I Ve(p) . Also notice that; by assumpti

1- Complete the following proof. Theorem Let S ER: Let p â‚¬ R The following two statements are equitlent: (i) Ve > Ve(p)o (S| {p}) "onemptu: (ii) P= lim dn for some sequence (an) contained in $| {p} (Recall that if any of the above equivalent conditions holds, then we say that is a limit poinf of$.) Proof: (ii) Let â‚¬ be given_ Qur goal , is to show that Ve(p)n(S| {p}) is nonempty: We have Uj p JN â‚¬N Vn >N dn â‚¬ Velp) In particular; aN+I Ve(p) . Also notice that; by assumption aN+1 â‚¬s {p}: This shows that Ve(p)n(s| {p}) is nonempty: (complete the proof. Hint: Notice that for all â‚¬N; by assumption; (p- #p+4) has nonempty intersection with {p} Construct sequence (dn) by letting On be point in the nonempty intersection of (p 2P+ and {p}- Use the Squeeze Theorem to show that indeed &r - 2- Let sequence of real numbers. Let s = {an : " â‚¬N} Prove that if L is limit point of (that is, if L â‚¬ $'), then there exists subsequence of (an) whose limit is L Proof: In the reading assignment posted to the course homework page we proved the following lemma: Lemma. Let E CR Let p â‚¬ E' and suppose that 0 is aJ open set containing Tlen On_ contains infinitely aiI pomts Since L â‚¬$', it follows from the above lemma that (L-LLtI)nshas infinitely many points 3n such that dnp (L-LL+I) L+ ins has infinitely many points 302` "g such that arz â‚¬ (L = ~L+ Lt nshas infinitely many points 3n3 nz such that Ors 3-L+ In ths way we obtain subsequence Un such that VkeN Sam < Lt} (complete the proof. Hint: Use the Squeeze Theorem to show that lim Ont = L)

#### Similar Solved Questions

##### Fiud the formula forwliere â‚¬â‚¬ and n =12_Using YOur conclusion of () t dexluce tormula tor{01520IymwlereR #tl(Hint; Euler # forula miglt he helpful. )
Fiud the formula for wliere â‚¬â‚¬ and n =12_ Using YOur conclusion of () t dexluce tormula tor {01520 Iym wlere R #tl (Hint; Euler # forula miglt he helpful. )...
##### Homework: 4.4 Homework Score: 0 of 1 pt 4.4.155 ofFind the area of the " region bounded by the graphs of the given equations. y=81x,y=x5,x=0,x=3The area is (Type an integer or simplified fraction )enier YAUI ansxerin the answer box &n0 {nen cick Check AnsitesAIL parts showing
Homework: 4.4 Homework Score: 0 of 1 pt 4.4.15 5 of Find the area of the " region bounded by the graphs of the given equations. y=81x,y=x5,x=0,x=3 The area is (Type an integer or simplified fraction ) enier YAUI ansxerin the answer box &n0 {nen cick Check Ansites AIL parts showing...
##### (15 pts) The function f is continuous and piece-wise linear on [0, 12]_The graph of f is shown below:y =f(x)Illustrate the midpoint Riemann sum of on [0, 12] for n = 3by sketching appropriate rectangles in the figure above(b) Compute the midpoint Riemann sum of f on [0, 12] for n = 3. Show your workExpress the limit of Riemann sums on [0, 12] as a definite integral:DO NOT EVALUATE THE INTEGRALIlim n-0O(vox)i + e* 12
(15 pts) The function f is continuous and piece-wise linear on [0, 12]_ The graph of f is shown below: y =f(x) Illustrate the midpoint Riemann sum of on [0, 12] for n = 3 by sketching appropriate rectangles in the figure above (b) Compute the midpoint Riemann sum of f on [0, 12] for n = 3. Show your...
##### Post this question to forumRecall that the "leading term ofa polynomial is the term containing the largest = ex Suppose f(c) 1525 + 132? + llc 17. Determine the leading term of15x^5PreviewSuppose g(z) 12.3 7r" + I 8. Determine the leading term of g.7x"7PreviewSuppose h(z) 6(3 8)( + 9)? . Determine the leading term of h; PreviewSubmit
Post this question to forum Recall that the "leading term ofa polynomial is the term containing the largest = ex Suppose f(c) 1525 + 132? + llc 17. Determine the leading term of 15x^5 Preview Suppose g(z) 12.3 7r" + I 8. Determine the leading term of g. 7x"7 Preview Suppose h(z) 6(3 8...
##### Dumann DaturDrono"Fen entrtnLeenemeeetiatnntata
Dumann Datur Drono" Fen entrtn Leenemeeetiatn ntata...
##### Iculate {neinct Wor< ouldul (In J) ora hci cnoinc {ollowing nath ARCDAEnc @ure Fhcrcm' ard10 < 10-JmJ(10 Nim )(10 'm )
iculate {neinct Wor< ouldul (In J) ora hci cnoinc {ollowing nath ARCDA Enc @ure Fhcrc m' ard 10 < 10-JmJ (10 Nim ) (10 'm )...
##### The legs of an isosceles right triangle increase in length at a rate of 2 m /s At what rate is the area of the triangle changing when the legs are m long? b At what rate is the area of the triangle changing when the hypotenuse At what rale long? the length of the hypotenuse changing? Write equation relaling ! the area of an isosceles right triangle, and the length of Ihe legs of the triangle,Differentiate both sides of the equation with respect toWhen the legs are mlong; the area of the triangle
The legs of an isosceles right triangle increase in length at a rate of 2 m /s At what rate is the area of the triangle changing when the legs are m long? b At what rate is the area of the triangle changing when the hypotenuse At what rale long? the length of the hypotenuse changing? Write equation ...
##### Determine tne ~Value(Round Intet dec mal placesnee tcu )Delermine the oulcone and concluslon Ihe teslAt Ine %"0 sionilic iC& level InerechduevidcilcyCClick select your answier(s)
Determine tne ~Value (Round Intet dec mal places nee tcu ) Delermine the oulcone and concluslon Ihe tesl At Ine %"0 sionilic iC& level Inere chduevidcilcy C Click select your answier(s)...
##### 18(D) 2 82) 36' +22)6+C Ji +8) (E) 2 4 %6' Vf +82+C1
18 (D) 2 82) 36' +22)6+C Ji +8) (E) 2 4 %6' Vf +82+C 1...
##### 5. [-/5 Points]DETAILSSCALCEIS iu.s.ui) Ui Zu subiiisjiuns OscFind the vertices and foci of the ellipse_ 9x2 16y2 32y 128vertices(xy)(smaller X-value)(x,Y) =(larger X-value)foci(x,Y)(smaller X-value)(larger X-valye)Sketch its graph.10
5. [-/5 Points] DETAILS SCALCEIS iu.s.ui) Ui Zu subiiisjiuns Osc Find the vertices and foci of the ellipse_ 9x2 16y2 32y 128 vertices (xy) (smaller X-value) (x,Y) = (larger X-value) foci (x,Y) (smaller X-value) (larger X-valye) Sketch its graph. 10...
##### Qucstion 16 Not yet answeredExpress the given quantity in terms . of sin X Or cosHatked Qut ofi0Ocos (37,*)Flag QUestlonSelect one cos~sin *cos *~cos
Qucstion 16 Not yet answered Express the given quantity in terms . of sin X Or cos Hatked Qut ofi0O cos (37,*) Flag QUestlon Select one cos ~sin * cos * ~cos...
##### The IQs of 700 applicants to certain college are approximately normally distributed with mean of 119 and standard deviation of 13. If the college requires an IQ of at least 100 , how many of these students will be rejected on this basis of IQ , regardless of their other qualifications" Note that IQs are recorded to the nearest integers_ Clickhere to viewpage of the standard normal distribution table Click here viekpage othe standard norma distribution tablestudent(s)(Round t0 the nearest wh
The IQs of 700 applicants to certain college are approximately normally distributed with mean of 119 and standard deviation of 13. If the college requires an IQ of at least 100 , how many of these students will be rejected on this basis of IQ , regardless of their other qualifications" Note tha...
##### Question 1: (5.0 marks) - Explain the term elemental analysis and state theapparatus used in performing qualitative analysis of organiccompound.Question 3: (5.0 marks)- Elucidate lassaigneâ€™s sodium fusion TestQuestion 4: (5.0 marks) (a) With appropriate diagram explainsolution, supernate, and suspension (b) Addition of Tollens reagentand Lucas Test to benzyl-aldehyde is used for identification ofwhat compound and state your observation
Question 1: (5.0 marks) - Explain the term elemental analysis and state the apparatus used in performing qualitative analysis of organic compound. Question 3: (5.0 marks) - Elucidate lassaigneâ€™s sodium fusion Test Question 4: (5.0 marks) ...
##### Compute the first-order partial derivatives.$$z= rac{x}{x-y}$$
Compute the first-order partial derivatives. $$z=\frac{x}{x-y}$$...
##### Experiment 15:the Law of Chemical Equilibrium and Le Chatelierâ€™sPrinciplePre-Lab QuestionsFor the reaction at 20áµ’C, NH3(aq) + H+(aq) â‡‹ NH4+(aq), the equilibrium constant iscalculated to be K = to 4.5 x108.Write the Equilibrium expression for this reaction. Keq =From the size of the number for Keq, does the equilibrium lieto the left or to the right?If the reaction between iron(III) ion and thiocyanate ion,Fe3+(aq)+ SCN-(aq) â‡‹ FeSCN2+(aq),yielded an equilibrium concentration of 0.10 M
Experiment 15: the Law of Chemical Equilibrium and Le Chatelierâ€™s Principle Pre-Lab Questions For the reaction at 20áµ’C, NH3(aq) + H+ (aq) â‡‹ NH4+(aq), the equilibrium constant is calculated to be K = to 4.5 x108. Write the Equilibrium expression for this reaction. Keq = From ...