How to prove something is closed and bounded, ie compact

[MeMoses]

Homework Statement

I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0).

Homework Equations

The Attempt at a Solution

How does one go about proving something is closed and bounded? My book is not very helpful and searching hasn't yielded much. This is only a part of the problem, but the rest should be doable once I get this.

 

[micromass]

Start by giving the definitions of "closed ball", "sphere", "closed" and "bounded".

 

[MeMoses]

Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started

 

[micromass]

Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started

What book are you using? Aren't those things in there?

 

[MeMoses]

Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?

 

[jbunniii]

Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?

That's not what bounded means. In fact, bounded has nothing to do with boundary. A bounded set is one that is fits inside a ball of some finite radius.

There are various equivalent definitions of closed. You must find out which one your book is using. Some possibilities:
* a set is closed if and only if its complement is open
* a set is closed if and only if it contains all of its limit points
* a set is closed if and only if it contains all of its boundary points