Let ECR. A point a E R is called a cluster point of E if E^(a−r, a+r)contains infinitely many points for every r > 0. Prove that everybounded infinite subset of R has at least one cluster point.

Given: E is bounded infinite subset of ℝ. To prove: E has at least one cluster point.

Answered: Let ECR. A point a E R is called 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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Lower bound. Infimum. Carefully read the definitions and remarks. Then attempt Question 2 at the end of the sub-section. Definition 0.1. Lower bound: Given a subset S C R. A real number a eR is a lower bound for the set ScR if a a. Remark 1. We write y = inf S S if the real number y is infimum or the greatest lower bound of a subset S C R. The Latin word infimum is used to denote the same quantity, abbreviated to inf: y = inf S. The subset s CR is called bounded from below if it has a lower bound. Remark 2. It follows from the Well-Ordering Principle that every set of real numbers that is nonempty and bounded from below has an infimum: As a matter of fact, a subset S C R is bounded from below iff the set S' := {x: -x € S} is bounded from above, and if S is nonempty and bounded from below, then – inf S' is the infimum of S. Remark 3. If the subset S c R has a smallest element, then inf S is simply the smallest element of S, often denoted min S. (For instance, if S is nonempty and…
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Let ECR. A point a E R is called a cluster point of E if En(a-r, a+r) contains infinitely many points for every r > 0. Prove that every bounded infinite subset of R has at least one cluster point.