*Proof*.
Write
The fact that we have nested intervals means

Thus, the form an increasing, bounded (above) sequence, while the form a decreasing, bounded (below) sequence.

By
a familiar fact
from calculus,
every
monotone, bounded sequence converges.
Thus, the
converge to some number
while the
converge to some number
which satisfy
.
Clearly, both *a* and *b* are elements of
,
because both are an elements of the closed interval
for any *n*. (Why?)
In fact, it's not hard to see that
is precisely the interval [*a*, *b*].
Finally, if
then we have *a* = *b*; that is,
= {*a*}.

Neal Carothers - carother@bgnet.bgsu.edu