Monotone Sequences

Proof. Suppose, for example, that is an increasing, bounded sequence; that is, suppose that we have

for some (fixed) number and all . Then, in particular, is a (nonempty) bounded set.

It follows that has a least upper bound (or supremum). In other words, there is an upper bound for the set, which we will again call , satisfying whenever  is any other upper bound for . We will show that converges to .

To this end, suppose that we are handed a small positive number , and consider the number . Since  < , we know that can't be an upper bound for . Thus, there is some  such that

Since our sequence is increasing, this means that we have

for all > . In particular,

for all > . Hence, converges to .

Finally, if we're given a decreasing, bounded sequence , just apply the first part of the proof to the increasing sequence .