How does one prove something is closed and bounded?

I'm wondering how one goes about proving something is both closed and bounded. I need to prove that a closed ball of radius r about x0 is closed and bounded. I'm just not sure how these proofs typically go, and I don't have a great definition of closed or bounded. Thanks for any help.

Re: How does one prove something is closed and bounded?

What are your definitions of closed and bounded?

Re: How does one prove something is closed and bounded?

There are a ton of ways to show a set is closed. Here are some:

1) Show the set contains all its limit points.

2) Show that the complement of the set is open.

3) Show it's a finite union, or arbitrary intersection, of other closed sets.

4) Show it equals its closure.

5) Show it's the pre-image of a closed set.

6) Show it's the image under a closed map of a closed set.

7) Perhaps it's known to be a compact subset of a Hausdorf space?

And I'm sure there are plenty more. Which to try depends on the circumstances.

If it's already been proven that the distance function in a metric space is a continuous function, then there's a very quick proof:

(Note that the distance function on R^n is obviously continuous, and so this works in R^n)

Let f = dist( *, x0):V -> R. Then D = f-inverse([0, r0]).

So since f is continous, and [0, r0] is known to be closed in R, have that D is closed.

I think that, maybe in this case, the next most simple way is to prove that D-complement is open.

Pick x not in D (that closed bal). Seek to show that x is in an open set that doesn't intersect D.

Since x not in D, dist(x, x0) > r. Let a = dist(x, x0) - r.

Let U = { x' | dist(x', x) < a/2}.

Then U open (why), x in U, and y in U implies y not in D (why? hint: triangle inequality).

So D complement is open. Therefore D is closed.

Re: How does one prove something is closed and bounded?

Quote:

Originally Posted by

**Shanter** I'm wondering how one goes about proving something is both closed and bounded. I need to prove that a closed ball of radius r about x0 is closed and bounded. I'm just not sure how these proofs typically go, and I don't have a great definition of closed or bounded. Thanks for any help.

Let $\displaystyle \mathcal{B}=\{x:d(x,x_0)\le r\}$. If $\displaystyle t\not\in\mathcal{B}$ then

$\displaystyle \alpha=d(t,x_0)-r>0.$

Now $\displaystyle \mathcal{O}=\{x:d(x,t)< \alpha\}$ is an open set such that $\displaystyle t\in\mathcal{O}~\&~\mathcal{O}\cap\mathcal{B}= \emptyset$.

That shows $\displaystyle \mathcal{B}^c$ is open. So $\displaystyle \mathcal{B}$ is closed.

If $\displaystyle u\in\mathcal{B}~\&~v\in\mathcal{B}$ show that $\displaystyle d(u,v)\le 2r$.