Proving Interior of S is NOT Empty: Baire's Theorem

[robertdeniro]

Homework Statement

let V be a vector space and S be a subset of V with the following property
for all v in V, there exist some positive integer p such that v/p is in S

given: S is closed and S has the property as described above. prove interior of S is NOT empty

Homework Equations

The Attempt at a Solution

since S is closed, we can define other closed sets S1, S2... such that the int(Sn)=0 and int(S U S1 U S2 U...)=/=0. once we have this we can apply baire's theorem.

problem is how do i define S1, S2,...

 

[lippyka]

? i don't think it is legal to define Sn={v/n|v in S} because every element of Sn is also in S, which would make S U S1 U S2 U...=S and int(S)=0