Sketch of a proof (you may have to fill in detail):Originally Posted byTexasGirl

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.

RonL