The real difficulty with the OP is the word

*nests*
There are cases where that concept is very general, see Kelly.

But you seem to restrict consideration to real numbers.

Saying that

is a nested sequence of sets usually means that if

then

.

If that is what

*nest* means then the sequence

is increasing, bounded above; the sequence

is decreasing , bounded below. In both cases the sequences converge and the limit(s) belong to the common part of the nest.

QUESTION: Do you mean something different by

nest?