Prove that the closure of a bounded set is bounded.

[dustbin]

Homework Statement

Prove that if S is a bounded subset of ℝ^n, then the closure of S is bounded.

Homework Equations

Definitions of bounded, closure, open balls, etc.

The Attempt at a Solution

See attached pdf.

 

[clamtrox]

That looks OK to me apart from a couple of typos.

There is an easier way to prove this too though. Do you know that the closure of S is the intersection of all closed subsets containing S?

 

[dustbin]

Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.

 

[Dick]

Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.

Your original proof looks fine to me. Except when you pick an epsilon, it should be "there exists epsilon" not "for all epsilon".