Proving Compactness of K in BR(0)

[Mathsgirl]

Homework Statement

R>0, let K be a closed subset of C such that K $$\subset$$ B_R(0) (so K is compact). Show that there exists 0 < r < R such that K$$\subset$$ Br(0).

Homework Equations

The Attempt at a Solution

Can I write B_R(0) = {x$$\in$$C : d(x,0) $$\leq$$R}?
I know that a compact set is closed and bounded.
Is it something to do with us using $$\subset$$ and not $$\subseteq$$?
As if it was $$\subseteq$$ then maybe K = B_R(0) then there wouldn't be an r. But as K is strictly contained in the ball there must be a bit of room for manoeuvre.

These are my thoughts on this problem. I'm not sure if they're correct or what the question is asking, and if they are I don't know how to write them more formally?

Thank you :)

 

[Dick]

Homework Statement

R>0, let K be a closed subset of C such that K $$\subset$$ B_R(0) (so K is compact). Show that there exists 0 < r < R such that K$$\subset$$ Br(0).

Homework Equations

The Attempt at a Solution

Can I write B_R(0) = {x$$\in$$C : d(x,0) $$\leq$$R}?
I know that a compact set is closed and bounded.
Is it something to do with us using $$\subset$$ and not $$\subseteq$$?
As if it was $$\subseteq$$ then maybe K = B_R(0) then there wouldn't be an r. But as K is strictly contained in the ball there must be a bit of room for manoeuvre.

These are my thoughts on this problem. I'm not sure if they're correct or what the question is asking, and if they are I don't know how to write them more formally?

Thank you :)

From the context of the problem, they must mean B_R(0) to be the open ball, d(x,0)<R. Not the closed ball. Otherwise, it wouldn't be true.

 

[Mathsgirl]

Oh that makes more sense.

I think I need to say something like you can always fit a smaller ball inside an open ball.

I remember in real analysis I showed for a non-empty bounded above set E and epsilon >0, there exists an x in E such that supE - epsilon < x $$\leq$$ sup E. Is it something like this?

 

[Dick]

Oh that makes more sense.

I think I need to say something like you can always fit a smaller ball inside an open ball.

I remember in real analysis I showed for a non-empty bounded above set E and epsilon >0, there exists an x in E such that supE - epsilon < x $$\leq$$ sup E. Is it something like this?

Something 'like' that could be made to work. But it's a lot easier if you use the definition of 'compact' and a covering of K by open balls.