1. Suppose that (a) and (b) are sequences with |a₁|≤ K |b₂| for all n, where K > 0,nand lim b =nn1x= 0. Prove directly from the definition of limit that lim a = 0.nn→∞

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Answered: 1. Suppose that (a) and (b) are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Problem 1RQ

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3. Suppose that (x,) is a bounded sequence and let T = {t€R:t < I, for infinitely many terms r,}. Prove that there exists a subsequence of (rn) converging to sup(T). Where precisely in your argument is the assumption of the boundedness of the sequence (r,) used? 4. Suppose that f,g : [0,1] [0, ac] are continuous functions and that
C. Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and converges to c (but never equal to c), we have limn → ∞ f(an) = ∞. (That is, prove a version of Relating Sequences to Functions for infinite limits. Make sure you handle both directions of the if and only if!)
(5) Consider the sequence defined by -( + 2) for n21 S1 = 1 Sn+1 = (Sn 4 (a) Prove by induction for all n E N (b) Prove the sequence is monotone. (c) Use the previous two parts to explain whether or not the limit of s, exists. If the limit exists, find the limit.
1. Let f: (0, 1)→R be uniformly continuous. Let (T) be a sequence in (0, 1). Suppose that lim, o In 0. Prove that lim, yo f(r) exists. %3D
5. Compute the marimum, minimum, in fimum and supremum (whenever these exist) for the following sets, (i) {1+ 1+(-1)" :n e N} (ii) {x : x E Q and x² 0 then lim, 0( 고) = 0 na+1
3. Prove the following statement. If (sn), (tn) are sequences of real numbers such that then (sn) is bounded and lim tn = 0, 818 lim Sntn = 0. 818
1. Which of the following statement (s) is/are TRUE? I: If lim an 0 then (In converges II: If 0 (In COllVerges. III: If an>0 and an (In D) II and III E) 1. II and II A) Only I B) Only II C) Only III
10. Consider f: (0, 2)→ R defined by f(x) = x". Assume that f has a limit at 0 and find that limit. (Hint: Choose a sequence (x)=1 converging to 0 such that the limit of the sequence {f(x)} is easy to determine.) 1
B. i. Prove directly from the definitions that if g: R → [1, ∞) is a function so that limx-c g(x) = L, then limx-c 1/g(x) = 1/L. ii. Prove the same result of the previous part, using Relating Sequences to Functions.
1. Let c > 0 be fixed and define the sequence {an} by setting an (an? + 3c) За,2 + с 2 а1 > 0, An+1 for n E N. Determine all a1 for which the sequence converges and in such a case find its limit. 2. Prove that if f is a bounded function on [0,1] satisfying f(ax) bf (x) for 0 1, then lim f(x) = f(0). | 1 3. Suppose that f:[1, c0) → R is uniformly continuous. Prove that there is a positive M such that- If (x)| 1. 4. A function f: [0,1] → R satisfies f (0) 0, and there exists a function g continuous on [0,1] and such that f + g is decreasing. Prove that the equation f (x) = O has a solution in the open interval (0,1).
number 1
5. lim(√n²+ n-n). Prove using the definition of limit

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1. Suppose that (a) and (b) are sequences with a ≤ K|b₂| for all n, where K > 0, and limb = = 0. Prove directly from the definition of limit that lim a = 0. n nx n→∞