1. Oct 8, 2009

1. The problem statement, all variables and given/known data

Theorem: If S is any bounded set in n space, and d>0 is given, then it is possible to choose a finite set of points pi in S such that every point p existing in S is within a distance d of at least one of the points p1, p2, ..., pm.

Prove this theorem assuming that the set S is both closed and bounded.

Prove this theorem, assuming only that S is bounded. [The difficulty lies in showing that the points pi can be chosen in S itself.

3. The attempt at a solution

Let S be a bounded set in n-space. By definition, there exists an M such that |p|< M for all p E S and S is a subset of B(0, M). Take po and p E S. ...

2. Oct 8, 2009

Dick

Use compactness?

3. Oct 8, 2009

HallsofIvy

Staff Emeritus
Well, that's the problem, Dick. It is easy if S is closed but suppose S is not closed?

You could, in that case, look at the closure of S but then you run into the problem that some of the finite number of points you get are boundary points of S that are not in S itself.

4. Oct 8, 2009

Dick

Oh yeah. Pick a point p0 in S. If B(p0,d) doesn't cover S, pick a point p1 outside the ball. If the union of B(p0,d)UB(p1,d) doesn't cover S, pick a point p2 outside the union. Continue. Doesn't that make the balls B(pk,d/2) disjoint? What can you conclude from that?

Last edited: Oct 8, 2009
5. Oct 8, 2009

Dick

The problem doesn't say that the set is bounded by d. It just says that it's bounded. d is a given number. Do what you did before and just say that there is an M such that |p|<M for all p in S.

6. Oct 8, 2009