(a)Prove that for every v = (x, y) e R² there exists a sequence vn =(xn, Yn) E R², where xn, Yn are rational numbers for alln e N, and such that limn→∞ Un = v.(b)Let (an)1 be a bounded sequence of real numbers. Define the set S = {x € R : x < an for infinitely many terms a,}.n=1Show that there exists a subsequence (ank)1 converging to s00sup S.%3D

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Description: (a)Prove that for every v = (x, y) e R² there exists a sequence vn =(xn, Yn) E R², where xn, Yn are rational numbers for alln e N, and such that limn→∞ Un = v.(b)Let (an)1 be a bounded sequence of real numbers. Define the set S = {x € R : x < an for

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Prove that for every v = (x, y) E R? there exists a sequence v, = (xn, Yn) € R2, where rn, Yn are rational numbers for all (a) n e N, and such that limn- Un = v.

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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