This looks suspiciously like the Baire category theorem (see the section headed "Proof" in that link).
This looks suspiciously like the Baire category theorem (see the section headed "Proof" in that link).
I actually used Baire category theorem. Here goes the solution:
Let .
Since , it follows that is a dense subset of F for all lambda. Hence each is a dense subset of F.
Suppose but . That means x is not an adherent point of . Therefore such that . But , so it must be true that .
On the other hand, since each is a dense subset of F, the category theorem guarantees that the countable intersection of these sets is a dense set in F. Thus , a contradiction.