Closed set, compact set, and a definition of distance between sets

[txy]

Homework Statement

Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows:

d(E,F) = inf_{x\in E , y\in F} | x - y |

(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 x_{k} in E and another sequence y_{k} in F such that as k increases, the distance between x_{k} and y_{k} 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.

 

[Office_Shredder]

For part (b), think about the graphs of 1/x² and -1/x² as an example.

That looks like a pretty good sketch argument. Only thing is that only a subsequence of each of those must converge, but it amounts to the same result here