MATH 202 A - Problem Set 4

Abstract

(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0) and let F ′ = F ∩B(r + α+ , x0) We have F ′ is a closed and bounded subset of R, thus it is compact. We also have

Cite this paper

@inproceedings{Krichene2012MATH2A, title={MATH 202 A - Problem Set 4}, author={Walid Krichene}, year={2012} }