Totally Bounded Sets: Proving Closure is Also Totally Bounded

[sazanda]

Homework Statement

Show that
If S is totally bounded in ℂ, then the S closure is also totally bounded in ℂ.

Homework Equations

The Attempt at a Solution

Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S let x be a limit points of S that is in S closure. but not in S.
hence x is in O (how can I show this?)
So x is in O for all x in S closure.
Hence S closure is totally bounded.

Am I on the right track?

 

[mighty2000]

Yes, you are on the right track. To show that x is in O, you can use the fact that x is a limit point of S and use the definition of a limit point to show that there exists a disc in O that contains x. Then, since x is in every disc in O, x is in O.