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Homework Statement
Let E and F be 2 non-empty subsets of [tex]R^{n}[/tex]. Define the distance between E and F as follows:
d(E,F) = [tex]inf_{x\in E , y\in F} | x - y |[/tex]
(a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E and F is a null set.
(b). If E and F (non-empty subsets of R^n) are compact sets and d(E,F) = 0, prove that the intersection of E and F cannot be the null set.
Homework Equations
The Attempt at a Solution
Part (b) of the question suggests that the closed sets in part (a) are not bounded. But I still can't find these 2 sets with d(E,F) = 0 and yet they don't intersect.
For part (b), maybe I can find a sequence of points [tex]x_{k}[/tex] in E and another sequence [tex]y_{k}[/tex] in F such that as k increases, the distance between [tex]x_{k}[/tex] and [tex]y_{k}[/tex] decreases. Then since E and F are compact, therefore these 2 sequences must converge. And since d(E,F) = 0, hence these 2 sequences must converge to the same point. And then since E and F are compact, this limit point must lie in both E and F, thus E and F share at least one common point.