Proof: Let be a generic element of
. To simplify the
notation, I will write

Let

and let

Then for

so

It follows by the th root rule (theorem 6.48) that . Hence it follows from theorem 7.6 that

Now

Here we have used the formula for a finite geometric series. Thus, from (7.8)

Now we want to calculate the limit appearing in the previous lemma. In order to do this it will be convenient to prove a few general limit theorems.

Proof: Suppose for all , and .

Case 1: Suppose . Then the conclusion clearly follows.

Case 2: Suppose
. Then by the formula for a geometric series

By the sum theorem and many applications of the product theorem we conclude that

Case 3: Suppose . Let . Then , so by Case 2 we get

Case 4: Suppose
where
and
.
Let
. Then

Now if we could show that , it would follow from this formula that

The next lemma shows that and completes the proof of theorem 7.10.

Proof: Let be a sequence of positive numbers such that .
Let
for each in
. We want to show that
. By the formula for a finite geometric series

so

Now

Since , we have , so by the squeezing rule , and hence

Proof: By lemma 7.7,

By theorem 7.10,

and putting these results together, we get

Proof: If

is a partition of , let

be the partition of obtained by multiplying the points of by . Then

If

is a sample for , let

be the corresponding sample for . Then

Let be a sequence of partitions of such that , and for each let be a sample for . It follows from (7.14) that . By the area theorem for monotonic functions (theorem 7.6), we have

Thus

Proof: The result for the case was proved in theorem 5.76. The case is done in the following exercise.

**Remark:** In the proof of lemma 7.7, we did not use the assumption
until line (7.9). For equation (7.9)
becomes

Since in this case ,we conclude that

This formula give us method of calculating logarithms by taking square roots. We know will be near to when is large, and can be calculated by taking successive square roots. On my calculator, I pressed the following sequence of keys

and got the result . My calculator also says that

. It appears that if I know how to calculate square roots, then I can calculate logarithms fairly easily.

Where in your proof do you use the fact that ?