Thread: Diameter of an Open Ball

I want to prove that for any ball , one has

So far I know the following:

. My problem is, what on earth does actually mean; how do I go about proving that the supremum is indeed less than or equal to 2r?

The statement seems obvious, but the proof for me is a little tricky because of this part. Thanks.

Originally Posted by FLT

I want to prove that for any ball , one has

So far I know the following:

. My problem is, what on earth does actually mean; how do I go about proving that the supremum is indeed less than or equal to 2r?

The statement seems obvious, but the proof for me is a little tricky because of this part. Thanks.

I think it must be , but then: :

this is also true for the supremum and we're done

Tonio

Thanks, I think it was a case of mis-applying the definition then!