2. In this problem, we will explore a different notion of convergence. Consider the collection O of sets given by: 0 = {A CR: A° is finite} U {0} U {R} (a) Prove that a countable union of sets in O is also in O. (b) Prove that a finite intersection of sets in O is also in O. (c) Given a sequence (r,) of real numbers and a real number L, we will say that (x„) O-converges to L, if for every element O e O, we have that LE O implies that there exists an N € N such that r, E O for all n > N. Prove that the familiar sequence () O-converges to 0.

2. In this problem, we will explore a different notion of convergence. Consider the collection O of sets given by: 0 = {A CR: A° is finite} U {0} U {R} (a) Prove that a countable union of sets in O is also in O. (b) Prove that a finite intersection of sets in O is also in O. (c) Given a sequence (r,) of real numbers and a real number L, we will say that (x„) O-converges to L, if for every element O e O, we have that LE O implies that there exists an N € N such that r, E O for all n > N. Prove that the familiar sequence () O-converges to 0.

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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