Topology(I included page from book that was said to follow idea as guide Theorem 5.10)Thankyou, I dont know how prove using sequences 1. Prove that the set of all irrational numbers is dense in R.Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences,show that sequence is dense in R, therefore conclude set of irrationals is dense in R union orroIC oboce Mnoiam5.10 Theorem: A metric space is second-countable if and only if it is separableProof: Suppose first that X is a separable metric space. Let {x,}j=1 be a dense 5.12sequence in X. Consider the family of open setsbns tibos YoB{B(x;1/n) : j > 1, n z 1}.%3DEXELet U be an open subset of X and let x e U. For some n 2 1, we have B(x;2/n) CU. Choose j so that d(xjx) < 1/n. Then x e heshows that B(x;1/n) C B(x;2/n) C U. Consequently B is a base of open sets. SinceB is countable, X is second-countable.Conversely, suppose that X is second-countable. Let {U-1 be a sequence ofopen sets in X that form a base. Let x, be any point in U,, n z 1. Then everynonempty open subset of X contains a point of the sequence {x}, so that the sequenceis dense in X, and X is separable. O€ B(x;1/n), and the triangle inequalityThe following theorem is the trickiest part of the circle of ideas covered in thissection.1gaiai doso bns Xax doso noi.Xlo isadue oldsin

We will use Theorem 5.10 along with the hint given in the question for the required proof.

1. Prove that the set of all irrational numbers is dense in R. Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences, show that sequence is dense in R, therefore conclude set of irrationals is dense in R

1. Prove that the set of all irrational numbers is dense in R. Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences, show that sequence is dense in R, therefore conclude set of irrationals is dense in R

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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