for all . By the comparison theorem for null sequences it follows that and are null sequences, and hence and

Then says ``for all '', so is true. Since is increasing, we have for all ,

By induction, we conclude that is true for all , i.e.

Proof: For all

an increasing sequence in , and suppose has an upper bound. Then converges. (Similarly, decreasing sequences that have lower bounds converge.)

Proof: Let be an upper bound for . We will construct a binary search sequence satisfying the following two conditions:

**i.** For every
, is an upper bound for ,

**ii.** For every
, is not an upper bound for .

Let

A straightforward induction argument shows that satisfies conditions i) and ii).

Let be the number such that . I will show that .

We know that
is a null sequence.
Let be a precision function for , so that for all
,

I will use to construct a precision function for .

Let . Since is not an upper bound for , there is a number such that . By condition i), I know that for all . Hence, since is increasing, we have for all :

Since we also have

Hence

This says that is a precision function for , and hence

We have for all . Suppose converges to a limit . Since for all , we can use the translation theorem to show that

so , and hence , so must be . Since for all , it follows from the inequality theorem that , and hence if converges, it must converge to . In order to show that converges, it is sufficient to show that is decreasing. (We've already noted that is a lower bound.)

Well,

so if I can show that for all , then I'll know that is decreasing. Now

I also note that , so I finally conclude that is decreasing, and hence . In fact, this sequence converges very fast, and is the basis for the square root algorithm used on most computers.

Claim: is a decreasing sequence.

Proof: For all
,

We will show by induction that (7.99) holds for all . Let

Then says , which is true since . For all ,

By induction, is true for all , and the claim is proved.

Let
Then
, since any precision function for
is also a precision function for
.
Hence

Thus , and hence . Since for all it follows from the inequality theorem that , and hence

We know that .

Hence

The snowflake was introduced by Helge von Koch(1870-1924)
who published his results
in 1906 [32]. Koch considered only the part of the boundary
corresponding to the bottom third of our polygon, which he introduced
as an example of a curve not having a tangent at any point.

The sequence from Exercise 7.82 is taken from [12, page 55, ex 20]