Old finals 364 NO SOL - 1 Let S be a subset of R which is...

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1. LetSbe a subset ofRwhich is bounded above and denoteα= supS.(a) Prove that for anyε >0, there existssSsuch thatα-ε < sα.(b) Prove that there exists a sequence{sn}n=1Ssuch thatsnα.2.(a) Prove thatf(x) = 2x+ 5 is uniformly continuous onR.(b) Suppose that a functionfis uniformly continuous on an intervalD.Let{xn}n=1be a Cauchy sequence inD. Prove that the sequence{f(xn}n=1is Cauchy.(c) Prove thatf(x) =1x4is not uniformly continuous on (0,1].(d) Prove thatf(x) =1x4is uniformly continuous on [1.).(e) TRUE OR FALSE: If a function is continuous on (0,3), then it isuniformly continuous on [1,2].Explain briefly or provide a coun-terexample.(f) TRUE OR FALSE: If a function is uniformly continuous on [0,3],then it is uniformly continuous on (1,2). Explain briefly or providea counterexample.3.(a) Suppose thatf:RRis continuous, and that{xn}n=1is a se-quence in the interval [a, b].Prove that ifxnconverges tox, thenx[a, b].(b) Suppose thatf:RRis continuous atc, and thatf(c)>0. Provethat there is a neighborhoodUofcsuch thatf(x)>0 for allxU.(c) Use the Intermediate Value Theorem to show that the functionf(x) =3x4+x-7 has a root in the interval [1,2].4.(a) Suppose thatSis an infinite subset ofR. Prove thatScontains asequence of distinct points.(b) Suppose thatSis a subset ofRwhich is unbounded above. ProvethatScontains a sequence{sn}n=1such thatsn→ ∞.(c) Suppose that the sequence{xn}n=1is unbounded above. Prove thatit has a subsequence{xnk}k=1such thatxnk→ ∞.5.(a) Suppose{xn}n=1is a sequence which does not converge tox. Provethat there existsε >0 and a subsequence{xnk}k=1such thatkxnk-xk ≥εfor allkinN.(b) Also show that there are at least two subsequences of{xn}n=1withdistinct limits.6. Use induction to prove that 12+ 22+. . .+n2=n(n+1)(2n+1)6.7. Letf:R → Rbe given byf(x) =x4sin(5x2+x3)ifx6= 00ifx= 0Prove thatfis differentiable atx= 0 and findf0(0).
8. Find the Taylor polynomialP3(x) for the functionf(x) =x1/2about thepointx= 1 and state the formula for the the remainder,R3(x).
1.(a) Prove thatlim sup(an+bn)lim supan+ lim supbnwhenever the right-hand-side is not of the form∞ − ∞or−∞+.(b) Give an example of sequences{an}and{bn}such that the aboveinequality is strict.2. Prove that iff:R → Ris continuous atc, thenfis bounded on someneighborhood ofc.3. Letf: [a, b]→ Rbe continuous. Prove thatfis bounded on [a, b].4. Decide whether the following four statements are TRUE or FALSE. Iftrue, EXPLAIN. If false, provide a COUNTEREXAMPLE.(a) All countable sets are well-ordered.(b) Forf:R→ R, continuity atcimplies continuity on a neighborhoodofc.(c) The functionxcosxhas a solution in the interval (0, π/2).(d) The sequence cosnhas a convergent subsequence.5. Letf:R → Rbe given byf(x) =x4sin(5x+2x3)ifx= 00ifx= 0Prove thatfis differentiable atx= 0 and findf(0).6. Prove thatf(x) = 1/xis not uniformly continuous on (0,1].7.(a) Find the Taylor polynomialP3(x) for the functionex2, withx0= 0.(b) State the Mean Value Theorem.

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