Sketch of a proof (you may have to fill in detail):Originally Posted by TexasGirl
1. There exist a such that for all and .
For if this were not the case we could construct a sequence with
limit , but this contradicts closed; as closed means the limit must be in .
(A closed set contains all its limit points)
2. Let be the union of all open balls centred on points in of diameter ,
and similarly for . Then and have the required properties.