Monotone Sequences

Theorem. spacer A monotone, bounded sequence of real numbers converges.


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

spacer

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

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

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

spacer

Since our sequence is increasing, this means that we have

spacer

for all spacer > spacer . In particular,

spacer

for all spacer > spacer . Hence, spacer converges to spacer .

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

Neal Carothers - carother@bgnet.bgsu.edu

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