Yes, I did read post 4 and it does make sense. Thank you for the clarification.

[cookiesyum]

Homework Statement

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 p_i in S such that every point p existing in S is within a distance d of at least one of the points p₁, p₂, ..., p_m.

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 p_i can be chosen in S itself.

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 p_o and p E S. ...

 

[Dick]

Use compactness?

 

[HallsofIvy]

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.

 

[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?

 

[Dick]

For the proof if S is closed, would the following be correct logic?

Let S be a closed and bounded set. By definition, there exists a d such that |p|< d for all pES and S is a subset of B(0,d) and Rn\S is an open set. Pick two points pES and p_oES. Then, |p-p_o| < d. Hence, every point pES is within d of at least one other point.

I feel like I'm missing something huge. I don't think I really understand the problem.

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.

 

[cookiesyum]

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.

I guess I don't know where to go after that though? Take p and p_oES. Say the distance between them is some d >0. Show that the distance between them and another point p_m is also d? But how?

 

[Dick]

I guess I don't know where to go after that though? Take p and p_oES. Say the distance between them is some d >0. Show that the distance between them and another point p_m is also d? But how?

Did you read post 4? Did it make sense?