Proof: We noted in theorem 5.40 and exercise 5.47 that

we conclude from the squeezing rule that

We also have by (5.43) that

so that

By (7.3) and the squeezing rule

and hence

Also,

and we call a

If , then

If then

is some number between and .

monotonic function from the interval to . Then for every sequence of partitions of such that , and for every sequence where is a sample for , we have

Proof: We will consider the case where is increasing. The case where
is
decreasing is similar.

For each partition and sample , we have for

Hence

i.e.,

By theorem 7.1 we have

and

so by the squeezing rule,