31.(a) Let A = {3+1.r ER. Show that A is a bounded subset of R.Hence find inf A and sup A.: 32x2+1%3D(b) For each a €R and Vb > 0, | a |= b if and only if a = ±b.Using this fact or otherwise find the set of all x ER that satisfythe equation | 2.x2 – 1 |=| x+ 2 |.(c) Consider the sequence (G. 2..4:). Give a rule for obtaining:).2: 5 :8: 111the general term an of the sequence.(d) Prove that Vn e N, <.(e) Let an be a sequence in R defined inductively as a1 = 1 andan+1 := Van + 2, V n > 1. Prove that an is convergent and find its limit.

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Answered: (a) Let A = { 1.r €R. Show that A is a…

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(a) Let A = { 1.r €R. Show that A is a bounded subset of R. %3D 2x2+1 Hence find inf A and sup A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

31.
(a) Let A = {3+1.r ER. Show that A is a bounded subset of R.
Hence find inf A and sup A.
: 3
2x2+1
%3D
(b) For each a €R and Vb > 0, | a |= b if and only if a = ±b.
Using this fact or otherwise find the set of all x ER that satisfy
the equation | 2.x2 – 1 |=| x+ 2 |.
(c) Consider the sequence (G. 2..4
:). Give a rule for obtaining
:).
2: 5 :8: 111
the general term an of the sequence.
(d) Prove that Vn e N, <.
(e) Let an be a sequence in R defined inductively as a1 = 1 and
an+1 := Van + 2, V n > 1. Prove that an is convergent and find its limit.

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