Consider a bounded, monotonic, real-valued function on the interval
. The define sequences
- Prove that
and that
- Prove that the two sequences
and
converge to
.
- State and prove a generalization of the above to interval
.
- Proof.First, we define two step functions,
where
denotes the greatest integer less than or equal to
. Then we define a partition of
,
For any
we have
So,
and
are constant on the open subintervals of the partition
.
Sinceis monotonically increasing and
for all
(by the definition of
and
) we have
- Proof. From part (a) we have
since
. Since
does not depend on
we have
Therefore,
- Claim: If
is a real-valued function that is monotonic increasing and bounded on the interval
, then
for
and
defined as follows:
Proof. Let
be a partition of the interval
. Then, define step functions
and
with
and
for
. By these definitions we have
for all
(since
is monotonic increasing). Since
is bounded and monotonic increasing it is integrable, and
And, since
, and
we have
You can prove (c) much more easily by defining a new function g(x)=f((b-a)x+a) and using parts (a) and (b) on g