Proof:

**Case 1:**- []. By the formula for
factoring (3.78),
we have for all
and all

so

If we let in this formula, we get

Since is a null sequence, it follows from the comparison theorem for null sequences that ; i.e., . **Case 2:**- [.] Let . Then , so by Case 1, . By the reciprocal theorem ; i.e., .

We have shown that the theorem holds in all cases.

Proof: The last assertion was shown in theorem 7.7, and the second
statement is clear, and it is also clear that
if .

Suppose that .
I will show that

Since is a null sequence, it follows from the comparison theorem for null sequences that is a null sequence, and then by the root theorem for null sequences (Theorem 7.19), it follows that is a null sequence.

To prove (7.65),
let be a precision function for
,
and let
.
Then
, so
,
so

.
and hence
, which is what we wanted to show.

Proof: We saw in theorem 3.71
that
for all
. If ,
. This sequence diverges, since it
is not bounded. If , then by the previous theorem
, so

Suppose now and . Then for all we have

Hence for all we have

By theorem 7.7, if and , then diverges, and hence is false for all ; i.e., diverges.

so informally, . Then is a convergent sequence, and

As an example, we have

- a)
- Let . Does converge? If it does, find in the form where .
- b)
- Let . Does converge? If it does, find in the form where
- c)
- Let . Does converge? If it does, find in the form where .

Snowflakes

area , and side . Note that an equilateral triangle with side has area . Starting with , we will now construct a sequence of polygons. will have sides, all having length . We let (so has sides of length ). To construct from we attach an equilateral triangle with side of length to the middle third of each side of .

A horizontal side
of might be replaced by
.
Each side of is replaced by 4
sides of length
, so will have
sides of length
. The figure
shows some of these polygons. I will call the polygons *snowflake
polygons*.
We have
for all . The *snowflake* is the
union of all of the sets ; i.e., a point is in if and only if it is in
for some
.

Find the area of (in terms of the area of ), for example

Then find the area of in terms of . Make any reasonable assumptions that you need. What is the perimeter of ?