**Problem 15.** Consider the following two properties:1. Every non-empty set that is bounded from above has a supremum.2. Every Cauchy sequence converges.Show that (2) implies (1). ((1) implies (2) was done in class, via the Bolzano-Weierstrass Theorem.)

As provided in the question, let us assume the following, 1. X be a non-empty set, which is bounded…

Answered: Problem 15. Consider the following two…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to…
4. Consider the sequence an = n > 1, k> 0. kn 2 (a) Determine the values of k for which the sequence converges. (b) Evaluate the limit of the sequence for the values of k where the limit exists. You must state if you use any standard limits, continuity or L’Hôpital's rule in your solution.
14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the same limit, say L. Show that (an) converges to L as
[(# Write out the first five terms of the sequence with, "1 ; n+2 determine whether the sequence converges, and if so find its limit. Enter the following information for an (器)” n n+4 n+2 aj = a2 = a3 = a4 = a5 = n + 4 lim n + 2 (Enter DNE if limit Does Not Exist.) 11 Does the sequence converge (Enter "yes" or "no").
#11

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Problem 15. Consider the following two properties: 1. Every non-empty set that is bounded from above has a supremum. 2. Every Cauchy sequence converges. Show that (2)=(1). ((1)→(2) was done in class, via the Bolzano-Weierstrass Theorem.)