I need help with #5. Thank you. 1. Prove or give counterexample.(a) If (xn) and (Yn) diverge, then (xn + Yn) diverges.(b) If (x„) and (Yn) diverge, then („Yn) diverges.(c) If (xn) and (Tn + Yn) converge, then (yn) converges.(d) If (x„) and („Yn) converge, then (yn) converges.COs n2. Use the definition of convergence to prove lim= 0n+0 n2 – n +13. Calculate the following limit analytically using theorems from the notes.lim (Vn² + n – n)4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.5. Prove that r € E' iff there exists a sequence (xn) in E\{x} such that x, → r.

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Answered: 5. Prove that r € E' iff there exists a…

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5. Prove that r € E' iff there exists a sequence (r„) in E\{r} such that r, → I.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

I need help with #5. Thank you.

1. Prove or give counterexample.
(a) If (xn) and (Yn) diverge, then (xn + Yn) diverges.
(b) If (x„) and (Yn) diverge, then („Yn) diverges.
(c) If (xn) and (Tn + Yn) converge, then (yn) converges.
(d) If (x„) and („Yn) converge, then (yn) converges.
COs n
2. Use the definition of convergence to prove lim
= 0
n+0 n2 – n +1
3. Calculate the following limit analytically using theorems from the notes.
lim (Vn² + n – n)
4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.
5. Prove that r € E' iff there exists a sequence (xn) in E\{x} such that x, → r.

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