The Nested Interval Theorem

Theorem. If is a sequence of nested, closed, bounded, nonempty intervals, then is nonempty. If, in addition, then consists of a single point.

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 [ab]. Finally, if then we have a = b; that is, = {a}.

Neal Carothers -