## Monotone Sequences

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

*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
.

Neal Carothers -
carother@bgnet.bgsu.edu